Exam 6: Counting

assume that you have 50 pennies and three jars, labeled A, B, and -In how many ways can you put the pennies in the jars, assuming that the pennies are distinguishable?

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

Make up a word problem in good English whose answer is $\left( \begin{array} { c } 15 \\4\end{array} \right) \cdot \left( \begin{array} { l } 7 \\3\end{array} \right)$

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

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 begin with 774?

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

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain exactly five elements, all of them even.

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 in position #483.

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 Ed and Gail are in the picture, standing next to each other?

How many permutations of the seven letters A, B, C, D, E, F, G do not have the vowels next to each other?

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

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

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

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?

assume that you have 50 pennies and three jars, labeled A, B, and -In how many ways can you put the pennies in the jars, assuming that the pennies are identical and each jar must have an even number of pennies put into it?

Each user has a password 6 characters long where each character is an uppercase letter, a lowercase letter, or a digit. Each password must contain at least one digit. How long will it take to check every possible character combination, if each check takes one unit of time.

How many truth tables are possible for compound propositions with the five variables p, q, r, s, t ?

How many permutations of 12345 are there that leave 3 in the third position but leave no other integer in its own position?

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

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