There are two bags each of which contains n balls. A man has to select an equal number of balls from both the bags. The number of ways in which a man can choose at least one ball from each bag is -
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No. of ways of selecting r balls from a bag containing n balls = n C r
∴ No. of ways of selecting r balls from each of two bags and each containing balls
= n C r × n C r = ( n C r ) 2 .
∴ No. of ways of choosing at least one ball from each bag
= ( n C 1 ) 2 + ( n C 2 ) 2 + .... + ( n C n ) 2
= [( n C 0 ) 2 + ( n C 1 ) 2 + ( n C 2 ) 2 + ..... + ( n C n ) 2 ] – ( n C 0 ) 2
= 2n C n – 1. [Using ( n C 0 ) 2 + ...... + ( n C n ) 2 = 2n C n ].
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