Mathemagic: combinations-possible sitting arrangements in a row

COMBINATIONS

PROBLEM:

 If there are 5 boys and 5 girls, all is to be seated on 10 seats in a row. Then what is the possible number of ways ,they can sit. Condition is that no two same gender sit together.

Solution:

1.)     If we start with a boy, then 1^st seat can be filled in 5 ways. And same like this, 2^nd seat also can be filled by a girl in 5 ways.

Next, 3^rd seat can be filled by a boy now in 4 ways, and 4^th seat also can be filled by a girl in 4 ways.

Proceeding in this manner,

We get possible number of ways = 5*5*4*4*3*3*2*2*1*1 = 14400 ways

2.)     if we start with a girl , then we get also

Possible number of ways = 14400 ways

Hence possible number of ways for this sitting arrangement will be = 14400+14400 = 28800 ways

General formula:

if there are n men and n women , all is to be seated on 2n seats in a row. Then what is the possible number of ways ,they can sit. Condition is that no two same gender sit together.

possible number of ways for this sitting arrangement will be,

                [n*n*(n-1)*(n-1)*(n-2)*(n-2)*…*1*1] + [n*n*(n-1)*(n-1)*(n-2)*(n-2)*…*1*1]

or,   = 2[n*n*(n-1)*(n-1)*(n-2)*(n-2)*…*1*1] = 2[n*(n-1)*(n-2)*…*1*1]² = 2(n!)²