Question

There are two bags can each of which contains `n` balls. A man has to select an equal number of balls from both the bags. Prove that the number of ways in which a man can choose at least one ball from each bag is `^2n C_n-1.`

Answer 1

Number of ways of selecting r balls from a bag containing n balls `= .^(n)C_(r)`
`:.` Number of ways of selecting r balls from each of two bags
`= ^(n)C_(r) xx .^(n)C_(r) = (.^(n)C_(r))^(2)`
`:.` Number of ways of selecting 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`