Question 1

consider all bit strings of length 12.
-How many have exactly four 1's and none of these 1's are adjacent to each other?

Accepted Answer

The answer of consider all bit strings of length 12.
-How...

Question 2

a club with 20 women and 17 men needs to form a committee of size six.
-How many committees are possible?

Accepted Answer

The answer of a club with 20 women and 17...

Question 3

How many permutations of the seven letters A, B, C, D, E, F, G have E in one of the ﬁrst two positions?

Accepted Answer

The answer of How many permutations of the seven letters...

Question 4

suppose that a "word" is any string of seven letters of the alphabet, with repeated letters allowed.
-How many words begin with A or end with B?

Accepted Answer

The answer of suppose that a "word" is any string...

Question 5

Find the next largest permutation in lexicographic order after 52143.

Accepted Answer

The answer of Find the next largest permutation in lexicographic...

Question 6

consider all bit strings of length 12.
-How many begin with 11 and end with 10?

Accepted Answer

The answer of consider all bit strings of length 12.
-How...

Question 7

A game consisting of ﬂipping a coin ends when the player gets two heads in a row, two tails in a row, or ﬂips the coin four times. (a) Draw a tree diagram to show the ways in which the game can end. (b) In how many ways can the game end?

Accepted Answer

The answer of A game consisting of ﬂipping a coin...

Question 8

suppose that a "word" is any string of seven letters of the alphabet, with repeated letters allowed.
-How many words end with the letter T?

Accepted Answer

The answer of suppose that a "word" is any string...

Question 9

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain exactly three elements.

Accepted Answer

The answer of Find the number of subsets of S...

Question 10

How many 5-element DNA sequences use all four bases, A, T, C, and G?

Accepted Answer

The answer of How many 5-element DNA sequences use all...

Question 11

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain neither 5 nor 6.

Accepted Answer

The answer of Find the number of subsets of S...

Question 12

How many permutations of the seven letters A, B, C, D, E, F, G do not have vowels on the ends?

Accepted Answer

The answer of How many permutations of the seven letters...

Question 13

How many permutations of the seven letters A, B, C, D, E, F, G neither begin nor end with A?

Accepted Answer

The answer of How many permutations of the seven letters...

Question 14

Find the number of permutations of the letters in your last name.

Accepted Answer

The answer of Find the number of permutations of the...

Question 15

let A be the set of all strings of decimal digits of length ﬁve. For example, 00312 and 19483
are strings in A.
-How many strings in A have exactly one 5?

Accepted Answer

The answer of let A be the set of all...

Question 16

Find the number of subsets of S = {1, 2, 3, . . . , 10} with exactly ﬁve elements, two of which are 3 and 4.

Accepted Answer

The answer of Find the number of subsets of S...

Question 17

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

Accepted Answer

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

Question 18

suppose that a "word" is any string of seven letters of the alphabet, with repeated letters allowed.
-How many words begin with A or B and end with A or B?

Accepted Answer

The answer of suppose that a "word" is any string...

Question 19

nine people (Ann, Ben, Cal, Dot, Ed, Fran, Gail, Hal, and Ida) are in a room. Five of them
stand in a row for a picture.
-In how many ways can this be done if Ben is to be in the picture?

Accepted Answer

The answer of nine people (Ann, Ben, Cal, Dot, Ed,...

Question 20

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain both 5 and 6.

Accepted Answer

The answer of Find the number of subsets of S...

Exam 6: A: Counting

consider all bit strings of length 12. -How many have exactly four 1's and none of these 1's are adjacent to each other?

a club with 20 women and 17 men needs to form a committee of size six. -How many committees are possible?

How many permutations of the seven letters A, B, C, D, E, F, G have E in one of the ﬁrst two positions?

suppose that a "word" is any string of seven letters of the alphabet, with repeated letters allowed. -How many words begin with A or end with B?

Find the next largest permutation in lexicographic order after 52143.

consider all bit strings of length 12. -How many begin with 11 and end with 10?

A game consisting of ﬂipping a coin ends when the player gets two heads in a row, two tails in a row, or ﬂips the coin four times. (a) Draw a tree diagram to show the ways in which the game can end. (b) In how many ways can the game end?

suppose that a "word" is any string of seven letters of the alphabet, with repeated letters allowed. -How many words end with the letter T?

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain exactly three elements.

How many 5-element DNA sequences use all four bases, A, T, C, and G?

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain neither 5 nor 6.

How many permutations of the seven letters A, B, C, D, E, F, G do not have vowels on the ends?

How many permutations of the seven letters A, B, C, D, E, F, G neither begin nor end with A?

Find the number of permutations of the letters in your last name.

let A be the set of all strings of decimal digits of length ﬁve. For example, 00312 and 19483 are strings in A. -How many strings in A have exactly one 5?

Find the number of subsets of S = {1, 2, 3, . . . , 10} with exactly ﬁve elements, two of which are 3 and 4.

suppose that a "word" is any string of seven letters of the alphabet, with repeated letters allowed. -How many words begin with A or B and end with A or B?

nine people (Ann, Ben, Cal, Dot, Ed, Fran, Gail, Hal, and Ida) are in a room. Five of them stand in a row for a picture. -In how many ways can this be done if Ben is to be in the picture?

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain both 5 and 6.