ENGG2430D: Engineering Mathematics IIISpring 2015

Problem Set Assignment 1 Due: January 23nd, 2015, 6pm

Prob. 1 (20pts) How many different linear arrangements are there of the letters A, B, C,D, E, F, G for which

(a) (5pts) A and B are next to each other?

(b) (5pts) A is before B?

(c) (5pts) G is not last in line?

(d) (5pts) If we add another letter C, which means there 8 letters of A, B, C, C, D,E, F, G. How many linear arrangements for which D is at least next to one C?(Hint: There are two C letters and they are treated as identical objects.)

Prob. 2 (10pts) Show that(nk

)= n

k

(n−1k−1

), ∀n = 1, 2, · · · , k = 1, 2, · · · , n.

Prob. 3 (30pts) In the lecture, we have learned the binomial theorem

(x + y)n =n∑

k=0

(n

k

)xkyn−k.

Now prove the following identities by utilizing this binomial theorem.

(a) (10pts)(2n0

)+(2n2

)+(2n4

)+ · · ·+

(2n2n

)= 22n−1, ∀n = 1, 2, 3, · · · .

(b) (10pts) 12(n1

)+ 22

(n2

)+ 32

(n3

)+ · · ·+ n2

(nn

)= (n2 + n)2n−2, ∀n = 2, 3, 4, · · · .

(c) (10pts)(m0

)(nr

)+(m1

)(n

r−1

)+(m2

)(n

r−2

)+· · ·+

(mr

)(n0

)=(m+nr

), ∀m,n = 1, 2, 3, · · · , r =

0, 1, 2, · · · ,m + n. We take convention that(nk

)= 0 if k > n.

[Note: One basic identity we have learned, i.e.,(

mr−1

)+(mr

)=(m+1r

)is a special

case of this identity where we can take n = 1.]

Prob. 4 (15pts) Suppose A and B are two events (not necessarily disjoint) with P (A) =0.3 and P (B) = 0.5. Please find the largest and smallest values for P (A ∪ B) andexplain under what conditions the values can be attained. What’s the new result ifthe parameters change to P (A) = 0.3 and P (B) = 0.9?

Prob. 5 (25pts) A subset A is included by set S means that every member in subset A isalso contained in set S. Given a set S = {1, 2, · · · , N},

(a) (5pts) subset. How many subsets A are included by set S ?

(b) (10pts) k-subset. How many k-subsets A are included by set S, where k-subsetimplies |A| = k ?

(c) (10pts) k-multiset. The notion of multiset (or bag) is a generalization of thenotion of a set in which members are allowed to appear more than once. Howmany k-multisets A are included by set S, where k-multiset implies |A| = k ?

2