Question 1

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.

Accepted Answer

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

Question 2

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

Accepted Answer

The answer of Use the binomial theorem to expand \[(...

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?

Accepted Answer

The answer of A computer randomly prints three-digit codes, with...

Question 4

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

Accepted Answer

The answer of Find the coefficient of \(x ^ {...

Question 5

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

Accepted Answer

The answer of Use the binomial theorem to prove the...

Question 6

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

Accepted Answer

The answer of Use the binomial theorem to expand \[\left(...

Question 7

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

Accepted Answer

The answer of Let \(s _ { 1 } ,...

Question 8

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

Accepted Answer

The answer of Use the binomial theorem to expand \[(...

Question 9

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.

Accepted Answer

The answer of Show that in a group of ten...

Question 10

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

Accepted Answer

The answer of Use the binomial theorem to prove the...

Question 11

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

Accepted Answer

The answer of Explain how the Pigeonhole Principle can be...

Question 12

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

Accepted Answer

The answer of Use the binomial theorem to expand \[\left(...

Question 13

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

Accepted Answer

The answer of Use the binomial theorem to prove the...

Question 14

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

Accepted Answer

The answer of Use the binomial theorem to expand \[(...

Question 15

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

Accepted Answer

The answer of Find the coefficient of \(x^ { 8...

Question 16

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

Accepted Answer

The answer of Use the binomial theorem to expand \[(...

Question 17

A professor teaching a Discrete Math course gives a multiple choice quiz that has ten questions, each with
four possible responses: a, b, c, d. What is the minimum number of students that must be in the professor's
class in order to guarantee that at least three answer sheets must be identical? (Assume that no answers are
left blank.)

Accepted Answer

The answer of A professor teaching a Discrete Math course...

Question 18

Aprocomputerve that someis programmedsubset musttohaprinve bteensubsetsprintedof at{1,least2, 3, 4t,wice.5} at random. If the computer prints 40 subsets,

Accepted Answer

The answer of Aprocomputerve that someis programmedsubset musttohaprinve bteensubsetsprintedof at{1,least2,...

Question 19

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).

Accepted Answer

The answer of You pick cards one at a time...

Question 20

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

Accepted Answer

The answer of Find the coefficient of \(x ^ {...

Quiz 6: Counting

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 $( 2 a + b ) ^ { 4 }$

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?

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

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

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

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: $\left( \begin{array} { l } 6 \\0\end{array} \right) + \left( \begin{array} { l } 6 \\1\end{array} \right) + \cdots + \left( \begin{array} { l } 6 \\6\end{array} \right) = 2 ^ { 6 }$

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 $\left( x ^ { 2 } + \frac { 1 } { x } \right) ^ { 7 }$

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

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

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

Aprocomputerve that someis programmedsubset musttohaprinve bteensubsetsprintedof at{1,least2, 3, 4t,wice.5} at random. If the computer prints 40 subsets,

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

Quiz 6: Counting

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 (2a+b)4( 2 a + b ) ^ { 4 }(2a+b)4

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?

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−3x)5\left( x - \frac { 3 } { x } \right) ^ { 5 }(x−x3​)5

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

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

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 (x2+1x)7\left( x ^ { 2 } + \frac { 1 } { x } \right) ^ { 7 }(x2+x1​)7

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

Find the coefficient of x8x^ { 8 }x8 in the expansion of (x2+2)13\left( x ^ { 2 } + 2 \right) ^ { 13 }(x2+2)13

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

Aprocomputerve that someis programmedsubset musttohaprinve bteensubsetsprintedof at{1,least2, 3, 4t,wice.5} at random. If the computer prints 40 subsets,

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

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

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

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

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

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

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

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

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

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

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