A dance class consists of 22 students

A dance class consists of 22 students, of which 10 are women and 12 are men. If 5 men and 5 women are to be chosen and then paired off, how many results are possible? (please show how you got the answer thanks!)

Answer

First, we know that there are (10 choose 5) x (12 choose 5) =252(792) = 199584 ways that will will account for the number ofways to choose the men and women.
Now, we wish to know how many ways there could put into pairs.For the first woman, there are 5 choices of men. For the secondwoman, there are 4 choices remaining. For the third woman, thereare 3 choices remaining. For the fourth woman, there are only 2choices remaining, and only 1 choie left for the last woman. Thus,there are 5x4x3x2x1 = 120 ways.
Thus,
in total, there are 199584×120 = 23950080 ways First, we know that there are (10 choose 5) x (12 choose 5) =252(792) = 199584 ways that will will account for the number ofways to choose the men and women.
Now, we wish to know how many ways there could put into pairs.For the first woman, there are 5 choices of men. For the secondwoman, there are 4 choices remaining. For the third woman, thereare 3 choices remaining. For the fourth woman, there are only 2choices remaining, and only 1 choie left for the last woman. Thus,there are 5x4x3x2x1 = 120 ways.
Thus,
in total, there are 199584×120 = 23950080 ways total number of ways for choosing 5 women out of 10 women = 10C5 total number of ways for choosing 5 men out of 12 men= 12 C5 If we select a woman from 5 women, she can dance with any man ,so she has 5 ways to pick his partner. but the second woman has only 4 chances to pick his partner as the first woman has already choosen one man and there are only 4 available and for third woman it will be 3 ways and for fourh women it will be 2 ways and for fifth woman it will be only 1 way as all other men are selected by other women so ,number of ways for selecting their respective partner is 5*4*3*2*1 ways Finally the total number of ways are 10C5 *12C5*5*4*3*2*1

∴number of ways =23950080 ways

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