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Suppose there's a row of 8 chairs, in how many ways a boy or a girl can sit so that they fill the row? For example: it could be 7 boys or 1 girl or 1 boy or 7 girls to complete the row.

I'm not sure if this requires permutation or combinations to solve.

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You’re going to have to clarify the question. Must there be exactly one of one sex or the other, or can there be any number of boys and any number of girls so long as the total is $8$? Also, do we distinguish one boy from another, or do we care only whether someone is a boy or a girl? –  Brian M. Scott Sep 21 '11 at 0:49
    
Here we only care if someone is boy or a girl and there can be any number of boys or any number of girls but only 8 (either boy or girl) gets to sit on 8 chairs. –  John Sep 21 '11 at 0:55

1 Answer 1

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You don’t actually need either permutations or combinations for this one. Notice that if you know where the girls are sitting, you know the whole arrangement, since every chair is filled. Thus, to count the number of possible arrangements, you need only count the number of different ways the girls could sit. Since there could be any number of girls from $0$ through $8$, the girls could sit in any subset of the $8$ chairs, from $\varnothing$, the empty subset $-$ no girls at all $-$ to the entire set of $8$ chairs $-$ no boys at all. A set with $n$ elements has $2^n$ subsets, so there are $2^8 = 256$ different sets of chairs that the girls could occupy, and therefore $256$ different arrangements of the $8$ girls and boys.

(If by any chance you’re required to have at least one of each sex, you have to rule out two of these arrangements, the one consisting of $8$ boys and the one consisting of $8$ girls; that of course leaves $254$ arrangements.)

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