Exam 6: A: Counting

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

List the derangements of 1, 2, 3, 4.

Find the coefficient of $x ^ { 5 } \text { in } \left( 2 + x ^ { 2 } \right) ^ { 12 } .$

You have a pile of 20 identical blank cards. On each card you draw a circle, a plus, or a square. How many piles of 20 cards are possible?

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

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

How many ways are there to select 6 students from a class of 25 to hold six different executive positions on

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?

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain no odd numbers.

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

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

Using the ordinary alphabet and allowing repeated letters, find the number of words of length 8 that begin and end with the same letter.

How many ways are there to select 6 students from a class of 25 to serve on a committee?

How many different strings can be made using all the letters in the word GOOGOL?

Find the largest coefficient in the expansion of $( x + 3 ) ^ { 5 }$

suppose that a "word" is any string of seven letters of the alphabet, with repeated letters allowed. -How many words begin with AAB in some order?

How many 8-element DNA sequences contain exactly two of the four bases?

A club with 20 women and 17 men needs to choose three different members to be president, vice president,and treasurer. (a) In how many ways is this possible? (b) In how many ways is this possible if women will be chosen as president and vice president and a man as treasurer?

Using the ordinary alphabet and allowing repeated letters, find the number of words of length 8 that have at least one C.

Using the ordinary alphabet and allowing repeated letters, find the number of words of length 8 that have exactly one B.