If five integers are chosen from the set (1,2,3,4,5,6,7,8), must there be at lea

Question

If five integers are chosen from the set (1,2,3,4,5,6,7,8), must there be at least two integers with the property that the larger minus the smaller is 2? Write an answer that would convince a good but skeptical fellow student who has learned the statement of the pigeonhole principle but not seen an application like this one. Either describe the pigeons, the pigeonholes, and how the pigeons get to the pigeonholes, or describe a function by giving its domain, co-domain, and how elements of the domain are related to elements of the co-domain. 6. Given any set of 30 integers, must there be two that have the same remainder when they are by 25? Write an answer that would convince a good but skeptical fellow student who has learned the statement of the pigeonhole principle but not seen an application like this one. Either describe the pigeons, the pigeonholes, and how the pigeons get to the pigeonholes, or describe a function by giving its domain, co-domain, and how elements of the domain are related to elements of the co-domain. A club has seven members. Three are to be chosen to go as a group to a national meeting (a) How many distinct groups of three can be chosen? (b) If the club contains four men and three women, how many distinct groups of three contain two men and one woman? (c) If the club contains four men and three women, how many distinct groups of three contain at most two men? (d) If the club contains four men and three women, how many distinct groups of three contain at least one woman? 8. A large pile of coins consists of pennies, nickels, dimes, and quarters (at least 20 of each). (a) How many different collections of 20 coins can be chosen? (b) How many different collections of 20 coins chosen at random will contain at least 3 coins of each type? 9 The binomial theorem etates that for any real numbers a and b, (a+b)" (e)an-kbk for any integer n 0.

Explanation / Answer

7:

(a)

Here order of selection is not important so number of ways of selecting three members out of 7 is

C(7,3) = 35

(b)

Number of ways of selecting 2 men out of 4 men and one woman out of three women is

C(4,2) * C(3,1) = 18

(c)

Number of ways of selecting 2 men out of 4 men and one woman out of three women is

C(4,2) * C(3,1) = 18

Number of ways of selecting 1 men out of 4 men and two women out of three women is

C(4,1) * C(3,2) = 12

Number of ways of selecting 0 man out of 4 men and 3 women out of three women is

C(4,0) * C(3,3) = 1

So number of ways of selecting at most 2 men is

18+12+1 = 31

(d)

Number of ways of selecting 0 women is

C(4,3) = 4

So number of ways of selecting at least one woman is

35 - 4 = 31

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