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EECS 210
Answers to Homework 8
Due: 5/3/12
Reminder: The final exam will be Tuesday May 8 at 7:30 a.m. in 302 Snow.
1. (3) You are buying a bouquet of flowers and get to choose from six different varieties. How
many ways are there to put together a bouquet with 15 flowers?
C(6 + 15 – 1, 15) = C(20, 15)
2. (2) How many distinct arrangements are there of the letters in the word RECURRENCE?
10!/(3!•3!•2!)
3. (3) In how many of the arrangements in problem 2 do the R’s occur in consecutive positions?
This is basically arranging one group of three R’s and 7 other letters. So, instead of
10 things to arrange we have 8 giving us 8!/(3!•2!). (We still have to divide by the
repeated E’s and C’s.)
4. Consider the powerball game (59 white balls and 35 red balls).
a. (2) How many tickets are there which do not contain any multiple of 3?
C(40, 5)•C(24, 1) 19 white balls and 11 red balls are multiples of 3
b. (2) How many ways are there to match exactly four white balls and not the powerball?
C(5, 4)•C(54, 1)•C(34, 1) (4 white matches, 1 white mismatch, 1 red mismatch.)
c. (3) How many tickets are there in which the powerball number matches one of the other
numbers?
Choose the powerball, which then gives you one of the white balls, so all that
remains is to choose the remaining four numbers out of the 58 that remain so the
answer is:
35•C(58, 4).
Note: the following approach won’t work—choose five
white balls and then chose one of the five to be the powerball number. This fails
because some of the white balls chosen may be larger than 35 and thus those are
eliminated as possible matching numbers.
4. (2) Suppose set A has ten elements. How many symmetric relations can be defined on A?
255 If you look at the set A  A in matrix form, there are 10 pairs on the diagonal and
45 above the diagonal giving us 55 pairs to choose from.
A symmetric relation
corresponds to a subset chosen from these 55 pairs. Every time you choose a pair
above the diagonal you must put the corresponding pair below the diagonal into the
relation to ensure it’s symmetric. For example, if you put (2, 4) into the subset then
you’ll need to put (4, 2) into the corresponding symmetric relation.
5. (3) Expand (2x + 4y)5. Give exact values for all coefficients.
C(5,0)(2x)5 + C(5,1)(2x)4(4y) + C(5,2)(2x)3(4y)2 + C(5,3)(2x)2(4y)3 + C(5,4)(2x)(4y)4 +
C(5,5)(4y)5 = 32 x5 + 320 x4y + 1280 x3y2 + 2560 x2y3 + 2560 xy4 + 1024 y5