Rosen 4.3
p324.
1. List all the permutations of {a, b, c}.
The 3*2*1 ways to arrange these 3 elements:
abc, acb, bac, bca, cab, cba
3.
How many permutations {a, b, c, d, e, f, g} end with a?
Count the permutations of {b, c, d, e, f, g} : 6*5*4*3*2*1 = 720
9.
How many ways can 12 horses finish “in the money” if all orders of finish are possible?
Calculate P(12, 3) = 12*11*10 = 1320
11.
How many bit strings of length ten contain:
a)
exactly four 1s?
We choose which four places contain those 1s : C(10, 4) = (10*9*8*7) / (4*3*2*1)
= 210
b)
at most four 1s?
We add C(10, 4) + C(10, 3) + C(10, 2) + C(10, 1) + C(10, 0) = 386
c)
at least four 1s?
We subtract from the total number strings those that have only 0, 1, 2 or 3 1s :
2
10
– [ C(10, 0) + C(10, 1) + C(10, 2) + C(10, 3) ] = 1024 – 1 – 10 – 45 – 120 = 848
d)
an equal number of 0s and 1s?
Calculate C(10, 5) = 252
15.
In how many ways can a
set
of five letters be selected from the English alphabet?
C(26, 5) = 65780
17.
How many subsets with more than two elements does a set with 100 elements have?
We subtract from the total number of subsets those that have 0, 1, or 2 elements :
2
100