There are two bags 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.`

A bag contains 5 black and 6 red balls. Determine the number of ways in which 2 black and 3 red balls can be selected from lot.

The total number of ways of selecting 3 balls from a bag containing 7 balls is

A student is allowed to select at most n books from a collection of (2n+1) books. If the total number of ways in which a student selects at least one book is 63. then n equals to -

There are 2 identical white balls, 3 identical red balls, and 4 green balls of different shades. Find the number of ways in which they can be arranged in a row so that at least one ball is separated from the balls of the same color.

There are 'n' different books and 'p' copies of each. The number of ways in which a selection can be made from them is

A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball from the bag is thrice that of drawing a red ball, then find the number of blue balls in the bag.

There are four balls of different colors and four boxes of colors same as those of the balls. Find the number of ways in which the balls, one in each box, could be placed in such a way that a ball does not go to box of its own color.

Fifteen identical balls have to be put in five different boxes. Each box can contain any number of balls. The total number of ways of putting the balls into the boxes so that each box contains at least two balls is equal to a. .^9C_5 b. .^10 C_5 c. .^6C_5 d. .^10 C_6

Let theta=(a_(1),a_(2),a_(3),...,a_(n)) be a given arrangement of n distinct objects a_(1),a_(2),a_(3),…,a_(n) . A derangement of theta is an arrangment of these n objects in which none of the objects occupies its original position. Let D_(n) be the number of derangements of the permutations theta . There are 5 different colour balls and 5 boxes of colours same as those of the balls. The number of ways in which one can place the balls into the boxes, one each in a box, so that no ball goes to a box of its own colour is