Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 673 — #681
15.11. References 673
Problem 15.73.
Give a combinatorial proof for this identity:
!
!
nX k C i
D k C n C 1
k k C 1
iD0
Hint: Let S i be the set of binary sequences with exactly n zeroes, k C1 ones, and
a total of exactly i occurrences of zeroes appearing before the rightmost occurrence
of a one.
Problem 15.74.
According to the Multinomial Theorem 15.6.5, .x 1 C x 2 C C x k / n can be
expressed as a sum of terms of the form
!
n
x r 1
1
r 1 ; r 2 ; : : : ; r xr 2
2 : : : xr k
k :
k
(a) How many terms are there in the sum?
(b) The sum of these multinomial coefficients has an easily expressed value:
X
r 1 Cr 2 CCr k Dn;
r i 2N
Give a combinatorial proof of this identity.
!
n
D k n (15.29)
r 1 ; r 2 ; : : : ; r k
Hint: How many terms are there when .x 1 C x 2 C C x k / n is expressed as a sum
of monomials in x i before terms with like powers of these variables are collected
together under a single coefficient?
Problem 15.75.
You want to choose a team of m people for your startup company from a pool of n
applicants, and from these m people you want to choose k to be the team managers.
You took a Math for Computer Science subject, so you know you can do this in
!
n
m
!
m
k