Question 1

Use the binomial theorem to expand $( 2 c - 3 d ) ^ { 4 }$

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Question 2

Rental cars belong to one of twelve classiﬁcations depending on size and type of car. Each car is also assigned one of six color categories. How many cars does a rental agency need to guarantee that there are at least two cars of the same classiﬁcation and color?

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Question 3

A computer randomly prints three-digit codes, with no repeated digits in any code (for example, 387, 072, 760). What is the minimum number of codes that must be printed in order to guarantee that at least six of the codes are identical?

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Question 4

Use the binomial theorem to expand $( a + 2 ) ^ { 6 }$

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Question 5

Use the binomial theorem to expand $( 2 a + b ) ^ { 4 }$

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Question 6

Use the binomial theorem to prove the following: $$3 ^ { 100 } = \left( \begin{array} { c } 
100 \
0
\end{array} \right) + \left( \begin{array} { c } 
100 \
1
\end{array} \right) \cdot 2 + \left( \begin{array} { c } 
100 \
2
\end{array} \right) \cdot 2 ^ { 2 } + \left( \begin{array} { c } 
100 \
3
\end{array} \right) \cdot 2 ^ { 3 } + \cdots + \left( \begin{array} { c } 
100 \
99
\end{array} \right) \cdot 2 ^ { 99 } + \left( \begin{array} { c } 
100 \
100
\end{array} \right) \cdot 2 ^ { 100 }$$

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Question 7

Use the binomial theorem to expand $\left( x ^ { 2 } + \frac { 1 } { x } \right) ^ { 7 }$

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Question 8

Find the coeﬃcient of $x ^ { 8 } \text { in the expansion of } \left( x ^ { 2 } + 2 \right) ^ { 13 }$

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Question 9

A computer is programmed to print subsets of {1, 2, 3, 4, 5} at random. If the computer prints 40 subsets, prove that some subset must have been printed at least twice.

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Question 10

Find the coefficient of $x ^ { 7 } y ^ { 5 }$ in the expansion of $( 3 x - y ) ^ { 12 }$

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Question 11

Let $S _ { 1 } , S _ { 2 } \ldots , S _ { 101 }$ be 101 bit strings of length at most  9 .  Prove that there exist two strings, $8$ and $S _ { j }$, where $i \neq j$, that contain the same number of 0 's and the same number of 1's. (For example, strings 001001 and 101000 contain the same number of 0 's and the same number of 1 's.)

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Question 12

Use the binomial theorem to expand $( x + y ) ^ { 5 }$

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Question 13

Show that in a group of ten people (where any two people are either friends or enemies) there are either three mutual friends or four mutual enemies.

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Question 14

Use the binomial theorem to prove the following: $\left( \begin{array} { l } 
6 \
0
\end{array} \right) + \left( \begin{array} { l } 
6 \
1
\end{array} \right) + \cdots + \left( \begin{array} { l } 
6 \
6
\end{array} \right) = 2 ^ { 6 } .$

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Question 15

You pick cards one at a time without replacement from an ordinary deck of 52 playing cards. What is the minimum number of cards you must pick in order to guarantee that you get (a) a pair (for example, two kings or two 5_S). (b) three of a kind (for example, three 7_S).

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Question 16

Find the coefficient of $x ^ { 5 } y ^ { 6 }$ in the expansion of $( 2 x - y ) ^ { 11 }$

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Question 17

A computer network consists of six computers. Each computer is directly connected to zero or more of the other computers. Show that there are at least two computers in the network that are directly connected to the same number of computers.

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The answer of A computer network consists of six computers....

Question 18

Use the binomial theorem to prove the following: $$\left( \begin{array} { c } 
100 \
0
\end{array} \right) + \left( \begin{array} { c } 
100 \
2
\end{array} \right) + \left( \begin{array} { c } 
100 \
4
\end{array} \right) + \left( \begin{array} { c } 
100 \
6
\end{array} \right) + \cdots + \left( \begin{array} { c } 
100 \
98
\end{array} \right) + \left( \begin{array} { c } 
100 \
100
\end{array} \right) = \left( \begin{array} { c } 
100 \
1
\end{array} \right) + \left( \begin{array} { c } 
100 \
3
\end{array} \right) + \left( \begin{array} { c } 
100 \
5
\end{array} \right) + \cdots + \left( \begin{array} { c } 
100 \
97
\end{array} \right) + \left( \begin{array} { c } 
100 \
99
\end{array} \right)$$

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Question 19

Use the binomial theorem to expand $\left( x - \frac { 3 } { x } \right) ^ { 5 }$

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Question 20

Explain how the Pigeonhole Principle can be used to show that among any 11 integers, at least two must have the same last digit.

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Quiz 6: A: Counting

Use the binomial theorem to expand $( 2 c - 3 d ) ^ { 4 }$

Rental cars belong to one of twelve classiﬁcations depending on size and type of car. Each car is also assigned one of six color categories. How many cars does a rental agency need to guarantee that there are at least two cars of the same classiﬁcation and color?

A computer randomly prints three-digit codes, with no repeated digits in any code (for example, 387, 072, 760). What is the minimum number of codes that must be printed in order to guarantee that at least six of the codes are identical?

Use the binomial theorem to expand $( a + 2 ) ^ { 6 }$

Use the binomial theorem to expand $( 2 a + b ) ^ { 4 }$

Use the binomial theorem to expand $\left( x ^ { 2 } + \frac { 1 } { x } \right) ^ { 7 }$

Find the coeﬃcient of $x ^ { 8 } \text { in the expansion of } \left( x ^ { 2 } + 2 \right) ^ { 13 }$

A computer is programmed to print subsets of {1, 2, 3, 4, 5} at random. If the computer prints 40 subsets, prove that some subset must have been printed at least twice.

Find the coefficient of $x ^ { 7 } y ^ { 5 }$ in the expansion of $( 3 x - y ) ^ { 12 }$

Use the binomial theorem to expand $( x + y ) ^ { 5 }$

Show that in a group of ten people (where any two people are either friends or enemies) there are either three mutual friends or four mutual enemies.

Use the binomial theorem to prove the following: $\left( \begin{array} { l } 6 \\0\end{array} \right) + \left( \begin{array} { l } 6 \\1\end{array} \right) + \cdots + \left( \begin{array} { l } 6 \\6\end{array} \right) = 2 ^ { 6 } .$

You pick cards one at a time without replacement from an ordinary deck of 52 playing cards. What is the minimum number of cards you must pick in order to guarantee that you get (a) a pair (for example, two kings or two 5_S). (b) three of a kind (for example, three 7_S).

Find the coefficient of $x ^ { 5 } y ^ { 6 }$ in the expansion of $( 2 x - y ) ^ { 11 }$

A computer network consists of six computers. Each computer is directly connected to zero or more of the other computers. Show that there are at least two computers in the network that are directly connected to the same number of computers.

Use the binomial theorem to expand $\left( x - \frac { 3 } { x } \right) ^ { 5 }$

Explain how the Pigeonhole Principle can be used to show that among any 11 integers, at least two must have the same last digit.

Quiz 6: A: Counting

Use the binomial theorem to expand (2c−3d)4( 2 c - 3 d ) ^ { 4 }(2c−3d)4

Rental cars belong to one of twelve classiﬁcations depending on size and type of car. Each car is also assigned one of six color categories. How many cars does a rental agency need to guarantee that there are at least two cars of the same classiﬁcation and color?

A computer randomly prints three-digit codes, with no repeated digits in any code (for example, 387, 072, 760). What is the minimum number of codes that must be printed in order to guarantee that at least six of the codes are identical?

Use the binomial theorem to expand (a+2)6( a + 2 ) ^ { 6 }(a+2)6

Use the binomial theorem to expand (2a+b)4( 2 a + b ) ^ { 4 }(2a+b)4

Use the binomial theorem to expand (x2+1x)7\left( x ^ { 2 } + \frac { 1 } { x } \right) ^ { 7 }(x2+x1​)7

Find the coeﬃcient of x8 in the expansion of (x2+2)13x ^ { 8 } \text { in the expansion of } \left( x ^ { 2 } + 2 \right) ^ { 13 }x8 in the expansion of (x2+2)13

A computer is programmed to print subsets of {1, 2, 3, 4, 5} at random. If the computer prints 40 subsets, prove that some subset must have been printed at least twice.

Find the coefficient of x7y5x ^ { 7 } y ^ { 5 }x7y5 in the expansion of (3x−y)12( 3 x - y ) ^ { 12 }(3x−y)12

Use the binomial theorem to expand (x+y)5( x + y ) ^ { 5 }(x+y)5

Show that in a group of ten people (where any two people are either friends or enemies) there are either three mutual friends or four mutual enemies.

Use the binomial theorem to prove the following: (60)+(61)+⋯+(66)=26.\left( \begin{array} { l } 6 \\0\end{array} \right) + \left( \begin{array} { l } 6 \\1\end{array} \right) + \cdots + \left( \begin{array} { l } 6 \\6\end{array} \right) = 2 ^ { 6 } .(60​)+(61​)+⋯+(66​)=26.

You pick cards one at a time without replacement from an ordinary deck of 52 playing cards. What is the minimum number of cards you must pick in order to guarantee that you get (a) a pair (for example, two kings or two 5S). (b) three of a kind (for example, three 7S).

Find the coefficient of x5y6x ^ { 5 } y ^ { 6 }x5y6 in the expansion of (2x−y)11( 2 x - y ) ^ { 11 }(2x−y)11

A computer network consists of six computers. Each computer is directly connected to zero or more of the other computers. Show that there are at least two computers in the network that are directly connected to the same number of computers.

Use the binomial theorem to expand (x−3x)5\left( x - \frac { 3 } { x } \right) ^ { 5 }(x−x3​)5

Explain how the Pigeonhole Principle can be used to show that among any 11 integers, at least two must have the same last digit.

Use the binomial theorem to expand $( 2 c - 3 d ) ^ { 4 }$

Use the binomial theorem to expand $( a + 2 ) ^ { 6 }$

Use the binomial theorem to expand $( 2 a + b ) ^ { 4 }$

Use the binomial theorem to expand $\left( x ^ { 2 } + \frac { 1 } { x } \right) ^ { 7 }$

Find the coeﬃcient of $x ^ { 8 } \text { in the expansion of } \left( x ^ { 2 } + 2 \right) ^ { 13 }$

Find the coefficient of $x ^ { 7 } y ^ { 5 }$ in the expansion of $( 3 x - y ) ^ { 12 }$

Use the binomial theorem to expand $( x + y ) ^ { 5 }$

Use the binomial theorem to prove the following: $\left( \begin{array} { l } 6 \\0\end{array} \right) + \left( \begin{array} { l } 6 \\1\end{array} \right) + \cdots + \left( \begin{array} { l } 6 \\6\end{array} \right) = 2 ^ { 6 } .$

You pick cards one at a time without replacement from an ordinary deck of 52 playing cards. What is the minimum number of cards you must pick in order to guarantee that you get (a) a pair (for example, two kings or two 5_S). (b) three of a kind (for example, three 7_S).

Find the coefficient of $x ^ { 5 } y ^ { 6 }$ in the expansion of $( 2 x - y ) ^ { 11 }$

Use the binomial theorem to expand $\left( x - \frac { 3 } { x } \right) ^ { 5 }$