163 Discrete Mathematics Review 2 Use the following to answer ...

4. Find the number of subsets of S = {1,2,3,…,10} that contain exactly five elements, the
sum of which is even.
Solution. If the sum of the elements of a subset is even then the subset contains 0, or 2 ,
or 4 odd numbers and respectively 5, or 3, or 1 even numbers. Applying the
multiplication principle we see that the total number of such subsets is
C(5,0) ⋅ C(5,5) + C(5, 2) ⋅ C(5,3) + C(5, 4) ⋅ C(5,1) = 1⋅ 1+ 10⋅ 10 + 5⋅ 5 = 126.
5. Find the coefficient of x 5 y 6 in the expansion of (2x − y) 11 .
5
Solution. The corresponding term in the expansion isC(11, 5)(2 x) ( −y)
6 whence the
5 6
5
coefficient of x y is C(11,5) ⋅ 2 = 462⋅ 32 = 14784
Use the following to answer question 6:
In the questions below assume that you have 50 pennies and three jars, labeled A, B, and C.
6. In how many ways can you put the pennies in the jars, assuming that the pennies are
identical and each jar must have at least two pennies put into it
Solution. If we put two pennies in each box we will have 44 identical
(indistinguishable) pennies to be distributed into 3 labeled (distinguishable) jars. The
number of ways to distribute n indistinguishable objects into k distinguishable boxes is
(see page 377 and Theorem 2 on page 373)
Cn ( + k−1,
k−
1)
In our case the number is C (46,2) = 1035.
Use the following to answer question 7:
In the questions below you pick a bit string from the set of all bit strings of length ten.
7. What is the probability that the bit string has more 0s than 1s
Solution. Let p1, p2,
p3be the probabilities to have more 0s than 1s, more 1s than 0s,
and the same number of 0s and 1s, respectively. Then p 1
= p 2
,
5
p3 = C(10,5)(1/ 2) (1/ 2) 5 , and p1+ p2 + p3 = 1. Therefore
10
1 − C(10,5)/2
p1
= = 0.376953125
2
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