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July 5th, 2004

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At a dance there are 20 single women and 30 single men. In how many ways can they form 15 male-female couples.?

At a dance there are 20 single women and 30 single men. In how many ways can they form 15 male-female couples to dance simultaneously?

On going through some of yesterdays question, I came across this starred question just few minutes back. Really it is an interesting question. As I have some reservations on the answers provided by two other members I am providing my explanation below. You may peruse the same and a feed back would be much welcome.

1) The question is forming a couple team to 'DANCE SIMULTANEOUSLY' from available members. Since it is simultaneous, the selection by combination or permutations etc., are not logic. As a women combining with a men, cannot combine with another man for the same dance simultaneously. Hence, it is only by elimination process, we have to form the teams. As such my arrangements worked out on this principle only.

2) Team-1:
'Selection of a women from 20' = C(20,1) = 20 and a selection of a man from 30 = C(30,1) = 30.
==> The no of ways team 1 can be formed = 20 x 30 = 600. [As it is only a team formation and not a sitting/standing arrangements, the question of permutation does not arise].

Team - 2:
After selecting, one woman and one man for Team-1, they cannot be again considered for team-2, as explained in point 1 above; Hence, here the no of teams could be formed = C(19,1) x C(29,1) = 551

Similarly proceeding above, Team - 3: 18 x 28 = 504; Team-4: 17 x 27 = 459 and so on.

Hence the total no of teams could be formed =

(20x30) + (19x29) + (18x28) + (17x27) + (16x26) + (15x25) + (14x24) + (13x23) + (12x22) + (11x21) + (10x20) + (9x19) + (8x18) + (7x17) + (6x16)

= 600 + 551 + 504 + 459 + 416 + 375 + 336 + 299 + 264 + 231 + 200 + 171 + 144 + 119 + 96 = 4765

Answer: Number of teams that could be formed = 4765.

Wish you are explained; have a nice time; thanks

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