91Ó°ÊÓ

Determine the number of integer solutions of \(x_{1}+x_{2}+x_{3}+x_{4}=32\), where a) \(x_{i} \geq 0, \quad 1 \leq i \leq 4\). b) \(x_{i}>0, \quad 1 \leq i \leq 4 .\) c) \(x_{1}, x_{2} \geq 5, x_{3}, x_{4} \geq 7\). d) \(x_{i} \geq 8, \quad 1 \leq i \leq 4\). e) \(x_{i} \geq-2, \quad 1 \leq i \leq 4 .\) f) \(x_{1}, x_{2}, x_{3}>0,0

A committee of 12 is to be selected from 10 men and 10 women. In how many ways can the selection be carried out if (a) there are no restrictions? (b) there must be six men and six women? (c) there must be an even number of women? (d) there must be more women than men? (e) there must be at least eight men?

In how many ways can a gambler draw five cards from a standard deck and get (a) a flush (five cards of the same suit)? (b) four aces? (c) four of a kind? (d) three aces and two jacks? (e) three aces and a pair? (f) a full house (three of a kind and a pair)? (g) three of a kind? (h) two pairs?

Matthew works as a computer operator at a small university. One evening he finds that 12 computer programs have been submitted earlier that day for batch processing. In how many ways can Matthew order the processing of these programs if (a) there are no restrictions? (b) he considers four of the programs higher in priority than the other eight and wants to process those four first? (c) he first separates the programs into four of top priority, five of lesser priority, and three of least priority, and he wishes to process the 12 programs in such a way that the top-priority programs are processed first and the three programs of least priority are processed last?

In how many ways can the letters in WONDERING be arranged with exactly two consecutive vowels?

Pamela has 15 different books. In how many ways can she place her books on two shelves so that there is at least one book on each shelf? (Consider the books in any arrangement to be stacked one next to the other, with the first book on each shelf at the left of the shelf.)

How many ways are there to pick a five-person basketball team from 12 possible players? How many selections include the weakest and the strongest players?

Two \(n\)-digit integers (leading zeros allowed) are considered equivalent if one is a rearrangement of the other. (For example, 12033, 20331, and 01332 are considered equivalent five-digit integers.) a) How many five-digit integers are not cquivalent? b) If the digits 1,3, and 7 can appear at most once, how many nonequivalent five-digit integers are there?

Mr. and Mrs. Richardson want to name their new daughter so that her initials (first, middle, and last) will be in alphabetical order with no repeated initial. How many such triples of initials can occur under these circumstances?

In how many ways can we distribute eight identical white balls into four distinct containers so that (a) no container is left empty? (b) the fourth container has an odd number of balls in it?