MATH 351 Fall 2009 Homework 1 Due: Wednesday, September 30

Again we proceed in a similar fashion and count the committees when Mr. X and Ms.
A serve ( together. To do this we first choose the 2 other men from the remaining ( 5,
5 7
or = 10 and then choose the other 2 women from the remaining 7 or = 21.
2)
2)
Thus the total number of ways in which they can serve together is the product of
these two numbers of 210. Now the total number of committees in which they do not
serve together is our total number of committees with no restrictions (1120) minus
the committees where they do serve together (210). Thus the answer is 910.
Problem 8. Let r, n, and m be positive integers, and let r < n and r < m. Then
the following identity holds:
( ) n + m
=
r
r∑
( ) ( ) n m
.
i r − i
i=0
Prove it combinatorially by considering a group of n men and m women, and determining
(in two different ways) how many groups of size r are possible.
To prove this we follow the suggestion and consider a group of n men and m women,
and count the number of committees of r people we can choose from the total (n+m)
people. Clearly one way to count this is the left hand side (LHS) of the equation we
wish to verify.
Now consider counting the same thing but breaking it into cases were we consider
the committees that have no men, exactly 1 man, exactly 2 men, ...., all r are men.
To do this consider the arbitrary case where there are i men in the( committee ) and r-i
n
women. Thus we must choose the i men from the total n men or and we must
( )
i
m
choose the r-i women from the total m women or Thus the total number of
r − i
committees ( ) ( ) with r people with i of them men and r-i of them women is the product,
n m
. Now since we want to count all the possible committees we must add
i r − i
all the cases where i = 0, i = 1, ....i = r. This is exactly the right hand side of the
equation we wish to verify.
Since both the left and the right hand side count the same thing, they must be equal.
Problem 9. In the expansion of (z + 8y) 7 , what is the coefficient of z 4 y 3 ?
6