fix typos in ch3

Author: oscarlevin

source/exercises/counting-combperm.ptx

@@ -9,8 +9,8 @@
     <statement>
       <p>
         How many triangles are there with vertices from the points shown below?
-        Note that we are not allowing degenerate triangles - ones with all three vertices on the same
-        line,
+        Note that we are not allowing degenerate triangles <ndash/> ones with all three vertices on the same
+        line <ndash/>
         but we do allow non-right triangles.
         Explain why your answer is correct.
       </p>

source/exercises/counting-conc.ptx

@@ -14,7 +14,7 @@
         For each of the following counting problems,
         say whether the answer is <m>{10\choose 4}</m>,
         <m>P(10,4)</m>, or neither.
-        If you answer is <q>neither,</q>
+        If your answer is <q>neither,</q>
         say what the answer should be instead.
 
         <ol>
@@ -42,7 +42,7 @@
           <li>
             <p>
               If you want to wear 4 of your 10 bow ties next week (Monday through Sunday),
-              how many ways can this be accomplished?
+              in how many ways can this be accomplished?
             </p>
           </li>
 
@@ -56,7 +56,7 @@
           <li>
             <p>
               If 10 students come to their professor's office but only 4 can fit at a time,
-              how different combinations of 4 students can see the prof first?
+              how many different combinations of 4 students can see the prof first?
             </p>
           </li>
 
@@ -68,7 +68,7 @@
 
           <li>
             <p>
-              How many ways can you make the word <q>cake</q>
+              In how many ways can you make the word <q>cake</q>
               from the first 10 letters of the alphabet?
             </p>
           </li>
@@ -83,13 +83,13 @@
             <p>
               If you have 10 kids
               (and live in a shoe)
-              and 4 types of cereal, how many ways can your kids eat breakfast?
+              and 4 types of cereal, in how many ways can your kids eat breakfast?
             </p>
           </li>
 
           <li>
             <p>
-              How many ways can you arrange exactly 4 ones in a string of 10 binary digits?
+              In how many ways can you arrange exactly 4 ones in a string of 10 binary digits?
             </p>
           </li>
 
@@ -124,15 +124,15 @@
 
           <li>
             <p>
-              Each of your 10 bow ties match 4 pairs of suspenders.
+              Each of your 10 bow ties matches 4 pairs of suspenders.
               How many outfits can you make?
             </p>
           </li>
 
           <li>
             <p>
               After the party, the 10 kids each choose one of 4 party-favors.
-              How many outcomes?
+              How many outcomes are there?
             </p>
           </li>
 
@@ -144,7 +144,7 @@
 
           <li>
             <p>
-              How many ways can you split up 11 kids into 5 named teams?
+              In how many ways can you split up 11 kids into 5 named teams?
             </p>
           </li>
 
@@ -173,7 +173,7 @@
           <li>
             <p>
               Out of the 10 breakfast cereals available, you want to have 4 bowls.
-              How many ways can you do this?
+              In how many ways can you do this?
             </p>
           </li>
 
@@ -240,7 +240,7 @@
               Actually, <q>k</q> is the 11th letter of the alphabet,
               so the answer is 0.
               If <q>k</q> was among the first 10 letters,
-              there would only be 1 way - write it down.
+              there would only be 1 way <ndash/> write it down.
             </p>
           </li>
 
@@ -258,13 +258,13 @@
             <p>
               Neither.
               Note that this could not be
-              <m>{10 \choose 4}</m> since the 10 things and 4 things are from different groups. <m>4^{10}</m>, assuming each kid eats one type of cerial.
+              <m>{10 \choose 4}</m> since the 10 things and 4 things are from different groups. <m>4^{10}</m>, assuming each kid eats one type of cereal.
             </p>
           </li>
 
           <li>
-            <m>{10 \choose 4}</m> - don't be fooled by the <q>arrange</q>
-            in there - you are picking 4 out of 10
+            <m>{10 \choose 4}</m>. Don't be fooled by the <q>arrange</q>
+            in there. You are picking 4 out of 10
             <em>spots</em> to put the 1's.
           </li>
 

source/exercises/counting-pascal.ptx

@@ -13,7 +13,7 @@
 <exercise>
   <introduction>
     <p>
-      Suppose you are counting lattice paths from <m>(0,0)</m> to <m>(4,3)</m>.  We know that the number of such paths is <m>\binom{4+3}{4} = \binom{7}{6} = 35</m>.  Here is another way to count these paths: consider the cases for where your first step in the <m>y</m>-direction is.  There are five different options here; compute the number of paths for each case.
+      Suppose you are counting lattice paths from <m>(0,0)</m> to <m>(4,3)</m>.  We know that the number of such paths is <m>\binom{4+3}{4} = \binom{7}{6} = 35</m>.  Here is another way to count these paths: Consider the cases for where your first step in the <m>y</m>-direction is.  There are five different options here; compute the number of paths for each case.
     </p>
   </introduction>
   <task>
@@ -52,7 +52,7 @@
   <exercise>
     <statement>
       <p>
-        Explain why the coefficient of <m>x^5y^3</m> the same as the coefficient of <m>x^3y^5</m> in the
+        Explain why the coefficient of <m>x^5y^3</m> is the same as the coefficient of <m>x^3y^5</m> in the
         expansion of <m>(x+y)^8</m>? </p>
     </statement>
   </exercise>
@@ -67,7 +67,7 @@
     <task>
       <statement>
         <p>
-          Use the Binomial Theorem to write out the expansion of <m>(x+y)^5</m>.
+          Use the binomial theorem to write out the expansion of <m>(x+y)^5</m>.
         </p>
       </statement>
     </task>
@@ -81,7 +81,7 @@
     <task>
       <statement>
         <p>
-          How does your previous answer relate to what you get when you apply the Binomial Theorem to <m>(x+y)^6</m>?
+          How does your previous answer relate to what you get when you apply the binomial theorem to <m>(x+y)^6</m>?
         </p>
       </statement>
     </task>

source/exercises/counting-probability.ptx

@@ -63,7 +63,7 @@
   <exercise>  
     <statement>
       <p>
-        A group of 10 friends each has a deck of cards that they shuffle thoroughly. Each friend draws a card from their deck.  What is the probability that at least one pair of friends draw a matching card? 
+        Each of 10 friends has a deck of cards that they shuffle thoroughly. Each friend draws a card from their deck.  What is the probability that at least one pair of friends draw a matching card? 
       </p>
     </statement>
     <hint>
@@ -76,15 +76,15 @@
   <exercise>
     <statement>
       <p>
-        How many people do you need to have in a room to have a 50% chance that at least two people share the same birthday (day of the year)?  Assume that all birthdays are equally likely, and that nobody is born on leap day (February 29th).
+        How many people do you need to have in a room to have a 50% chance that at least two people share the same birthday (day of the year)?  Assume that all birthdays are equally likely, and that nobody is born on Leap Day (February 29th).
       </p>
     </statement>
   </exercise>
 
 <exercise>
   <statement>
     <p>
-      At your 20th high school reunion, you meet an old friend you hadn't heard from in years.  You talk about pets, specifically cats and dogs.  She tells you that she has two pets, and that at least one of them is a cat.  What is the probability that she has two cats?  (Assume that having a cat or a dog are equally likely.)
+      At your 20th high school reunion, you meet an old friend you hadn't heard from in years.  You talk about pets, specifically cats and dogs.  She tells you that she has two pets, and that at least one of them is a cat.  What is the probability that she has two cats?  (Assume that having a cat or a dog is equally likely.)
     </p>
   </statement>
 </exercise>
@@ -101,7 +101,7 @@
 <exercise>
   <statement>
     <p>
-      You are playing a shell game with three cups.  Under one cup are two green balls, under another cup are two red balls, and under the third cup are one green and one red ball.  You close your eyes and your friend rearranges the cups.  You then open your eyes and pick a cup at random.  You see that it contains a green ball.  What is the probability that the other ball under that cup is also green?  Explain your answer in terms of conditional probability.
+      You are playing a shell game with three cups.  Under one cup are two green balls, under another cup are two red balls, and under the third cup are one green and one red ball.  You close your eyes, and your friend rearranges the cups.  You then open your eyes and pick a cup at random.  You see that it contains a green ball.  What is the probability that the other ball under that cup is also green?  Explain your answer in terms of conditional probability.
     </p>
   </statement>
 </exercise>

source/exercises/counting-proofs.ptx

@@ -62,7 +62,7 @@
             <p>
               Use your answers to parts (a) and (b) to give a combinatorial proof of the identity
               <me>
-                {x+y \choose 2} - {x \choose 2} - {y \choose 2} = xy.
+                {x+y \choose 2} - {x \choose 2} - {y \choose 2} = xy
               </me>.
 
             </p>
@@ -141,7 +141,7 @@
         <p>
           Consider all the triangles you can create using the points shown below as vertices.
           Note that we are not allowing degenerate triangles
-          (ones with all three vertices on the same line)
+          (ones with all three vertices on the same line),
           but we do allow non-right triangles.
         </p>
 
@@ -185,7 +185,7 @@
             <li>
               <p>
                 State a binomial identity that your two answers above establish
-                (that is, give the binomial identity that your two answers a proof for).
+                (that is, give the binomial identity that your two answers are a proof for).
                 Then generalize this using <m>m</m>'s and <m>n</m>'s.
               </p>
             </li>
@@ -469,15 +469,15 @@
           <li>
             <p>
               How many ternary digit strings contain exactly <m>n</m> digits and <m>n-1</m> 2's. (Hint:
-              where can you put the non-2 digit,
+              Where can you put the non-2 digit,
               and then what could it be?)
             </p>
           </li>
 
           <li>
             <p>
               How many ternary digit strings contain exactly <m>n</m> digits and <m>n-2</m> 2's. (Hint:
-              see previous hint)
+              See previous hint.)
             </p>
           </li>
 
@@ -490,7 +490,7 @@
           <li>
             <p>
               How many ternary digit strings contain exactly <m>n</m> digits and no 2's. (Hint:
-              what kind of a string is this?)
+              What kind of a string is this?)
             </p>
           </li>
 
@@ -601,7 +601,7 @@
             </li>
 
             <li>
-              <p> Generalize. What if the rules changed and you played a best of <m>9</m> tournament (5 wins required)? What if you played an <m>n</m> game tournament with <m>k</m> wins required to be named champion? </p>
+              <p> Generalize. What if the rules changed, and you played a best of <m>9</m> tournament (5 wins required)? What if you played an <m>n</m> game tournament with <m>k</m> wins required to be named champion? </p>
             </li>
           </ol>
       </p>

source/practice/counting-probability.ptx

@@ -320,7 +320,7 @@
       <task>
         <statement>
           <p>
-            Compare the probability of getting two tails when flipping a coin twice to the probability of drawing a two red cards (without replacement).
+            Compare the probability of getting two tails when flipping a coin twice to the probability of drawing two red cards (without replacement).
           </p>
           <p>
             <m>P(\text{2 tails}) =</m><var name="Compute(1/4)" width="10" />; <m>P(\text{2 reds}) = </m> <var name="Compute((4*3)/(8*7))" width="10" />.  Difference: <var name="Compute(1/28)" width="10" />
@@ -330,7 +330,7 @@
       <task>
         <statement>
           <p>
-            Compare the probability of getting three tails when flipping a coin thrice to the probability of drawing a three red cards (without replacement).
+            Compare the probability of getting three tails when flipping a coin thrice to the probability of drawing three red cards (without replacement).
           </p>
           <p>
             <m>P(\text{3 tails}) =</m><var name="Compute(1/8)" width="10" />; <m>P(\text{3 reds}) = </m> <var name="Compute((4*3*2)/(8*7*6))" width="10" />.  Difference: <var name="Compute(3/56)" width="10" />
@@ -340,7 +340,7 @@
       <task>
         <statement>
           <p>
-            Compare the probability of getting four tails when flipping a coin four times to the probability of drawing a four red cards (without replacement).
+            Compare the probability of getting four tails when flipping a coin four times to the probability of drawing four red cards (without replacement).
           </p>
           <p>
             <m>P(\text{4 tails}) =</m><var name="Compute(1/16)" width="10" />; <m>P(\text{4 reds}) = </m> <var name="Compute((4*3*2*1)/(8*7*6*5))" width="10" />.  Difference: <var name="Compute(27/560)" width="10" />
@@ -440,7 +440,7 @@
           <var name="'6/(6^3)'" width="5" />
         </p>
         <p>
-          Is it possible for the dice to not all three show different numbers and not all three show the same number?  If not, the probability of this happening would be 0.  What is the probability?
+          Is it possible for the dice to show neither all the same nor all different numbers?  If so, the probability of this happening would be 0.  What is the probability?
         </p>
         <p>
           <var name="'1-((6*5*4)/(6^3)+6/(6^3))'" width="5" />

source/practice/counting-proofs.ptx

@@ -11,8 +11,8 @@
     </statement>
     <blocks>
       <block order="1">
-        <choice correct="yes"><p>Consider the question <q>How many two-digit numbers start with a 3 or 4?</q></p></choice>
-        <choice><p>Consider the question <q>How many 2-topping pizzas can you make choosing from 10 toppings?</q></p></choice>
+        <choice correct="yes"><p>Consider the question, <q>How many two-digit numbers start with a 3 or 4?</q></p></choice>
+        <choice><p>Consider the question, <q>How many 2-topping pizzas can you make choosing from 10 toppings?</q></p></choice>
       </block>
       <block order="2">
           <p>The first way to answer this is <m>10+10</m>.</p>