Given Twenty Couples how many different three member committees can be fored that do not contain both members of any of these couples

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- Jan 24th 2007, 06:20 AMRimasnumber of combinations
Given Twenty Couples how many different three member committees can be fored that do not contain both members of any of these couples

- Jan 24th 2007, 07:45 AMSoroban
Hello, Rimas!

Quote:

Given twenty couples, how many different 3-member committees can be formed

that do not contain both members of any of these couples?

There are 40 people available.

The first member $\displaystyle (A)$ can be any of the 40 people.

Of the remaining 39 people, the second member $\displaystyle (B)$ has 38 choices.

. . (He/she must not be A's spouse.)

Of the remaining 38 people, the third member $\displaystyle (C)$ has 36 choices.

. . (He/she must not be A's spouse or B's spouse.)

Hence, there are: .$\displaystyle 40 \times 38 \times 36 \:=\:54,720$ ways.

But these selections include an*ordering*of the members.

That is, $\displaystyle \{A,B,C\}$ is considered a different committe from $\displaystyle \{B,C,A\}$

Since three members can be ordered in $\displaystyle 3! = 6$ ways,

. . we divide by 6 to eliminate the duplications.

Answer: .$\displaystyle \frac{54,720}{6}\:=\:9,120$ ways.