hw.01
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ENGG2430D: Engineering Mathematics III Spring 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 ( n k ) = n k ( n-1 k-1 ) , ∀n =1, 2, ··· ,k =1, 2, ··· ,n. Prob. 3 (30pts) In the lecture, we have learned the binomial theorem (x + y) n = n X k=0 n k x k y n-k . Now prove the following identities by utilizing this binomial theorem. (a) (10pts) ( 2n 0 ) + ( 2n 2 ) + ( 2n 4 ) + ··· + ( 2n 2n ) =2 2n-1 , ∀n =1, 2, 3, ··· . (b) (10pts) 1 2 ( n 1 ) +2 2 ( n 2 ) +3 2 ( n 3 ) + ··· + n 2 ( n n ) =(n 2 + n)2 n-2 , ∀n =2, 3, 4, ··· . (c) (10pts) ( m 0 )( n r ) + ( m 1 )( n r-1 ) + ( m 2 )( n r-2 ) +···+ ( m r )( n 0 ) = ( m+n r ) , ∀m, n =1, 2, 3, ··· ,r = 0, 1, 2, ··· ,m + n. We take convention that ( n k ) = 0 if k>n. [Note: One basic identity we have learned, i.e., ( m r-1 ) + ( m r ) = ( m+1 r ) 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) and explain under what conditions the values can be attained. What’s the new result if the 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 is also contained in set S . Given a set S = {1, 2, ··· ,N }, (a) (5pts) subset. How many subsets A are included by set S ?
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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 ?
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