A bag contains 23 balls, of which 7 are identical. The number of ways of selecting 12 balls from the bag is....

Step-by-step solution

Total number of balls = 23 Number of identical balls = 7 Number of distinct balls = 23 - 7 = 16 We need to select 12 balls. Let the number of identical balls selected be k , where 0 k 7 . The remaining 12 - k balls must be selected from the 16 distinct balls. The number of ways to do this is 16 12-k . The total number of ways to select 12 balls is the sum of these possibilities for k = 0 to 7 : S = _ k=0 ⁷ 16 12-k S = 16 12 + 16 11 + 16 10 + 16 9 + 16 8 + 16 7 + 16 6 + 16 5 Using the property n r = n n-r , we can r