possible solutions 
1. 
A mathematics educator is planning to survey various professionals. Surveys will be sent to 44 biologists, 29 chemists, 37 physicists, 51 endocrinologists, and 22 epidemiologists. How many responses must the mathematics educator receive in order to guarantee that there will be at least 20 responses from the same professional group? 

2. 
Frank intends to invest his life savings of $200,000 in five mutual funds from a list of 20 such funds prepared by his mother, an investment banker. (a) How many different investments are possible if Frank invests $40,000 in each fund? (b) How many different investments are possible if Frank invests $20,000 in each of two funds, $40,000 in a third, $60,000 in each of two remaining funds? 

3. 
Six men and seven women, all of different heights, stand in a waiting line at the bookstore. (a) How many arrangements of these people are possible if the men stand in succession? (b) How many arrangements of these people are possible if two women stand at the very front of the line and two men are at the very end of the line? (c) How many arrangements of these people are possible if the men and women must alternate positions within the line? 

4. 
One definition for the word abracadabra is "a charm or incantation." How many unique arrangements are there for the letters in the word abracadabra? 

5. 
Two rooks on a chessboard attack each other if they are on the same row or the same column. Determine the number of ways in which 8 nonattacking identical rooks can be placed on the chessboard shown here. Squares with an "X" indicate forbidden positions that cannot be occupied. 

6. 
To complete this problem: Do ONE of the two problems listed under (I) below, OR Do BOTH problems listed under (II) below. I: (a) Prove that C(1,1) + C(2,1) + ... + C(n,1) = C(n + 1,2) for every positive integer n. (b) Prove that 1(1!) + 2(2!) + ... + n(n!) = (n + 1)!  1 for every positive integer n. II: (a) Prove that r*C(n,r) = n*C(n  1,r  1) for n >= r >= 1. (b) Prove that C(n,m)*C(m,k) = C(n,k)*C(n  k,m  k) for k <= m <= n. 
6Ia: 6Ib: 6IIa: 6IIb: 