Exam 6: Counting

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

If the permutations of 1,2,3,4,5,6 are written in lexicographic order, with 123456 in position #1, 123465 in position #2, etc., find the permutation immediately before 534126 .

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

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

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

In questions a club with 20 women and 17 men needs to form a committee of size six. -How many committees are possible if the committee must have three women and three men?

Find the number of permutations of the letters in the word COEFFICIENT.

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?

How many subsets with more than two elements does a set with 100 elements have?

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.)

Show that if five points are picked on or in the interior of a square of side length 2, then there are at least two of these points no farther than √2 apart.

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

You have 50 of each of the following kinds of jellybeans: red, orange, green, yellow. The jellybeans of each color are identical. (a) In how many ways can you put all the jellybeans in a row? (b) How many handfuls of 12 are possible? C.

Using the ordinary alphabet and allowing repeated letters, find the number of words of length 8.

Here is an incorrect solution to a problem. Find the error, explain why it is not correct, and give the correct answer. "Problem: Find the number of ways to get two pairs of two different ranks (such as 2 jacks and 2 fives) in a 4-card hand from an ordinary deck of 52 cards."

In questions 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 Hal or Ida (but not both) are in the picture?

How many permutations of the seven letters A, B, C, D, E, F, G have A immediately to the left of E?

How many 8-element DNA sequences start with C and end with C?

In questions a club with 20 women and 17 men needs to form a committee of size six. -How many committees are possible if the committee must have at least two men?

How many permutations of the seven letters A, B, C, D, E, F, G are there?