Question

A set of positive integers {1, 2 , 3 , 4, 5 , 6 , 7 , 8 , 9} is used to form a pack of nine cards. each card displays one positive integer without repetition from this set. grace wishes to select four cards at random from this pack of nine cards. (a) find the number of selections grace could make if the largest integer drawn among the four cards is either a 5, a 6 or a 7.

Answer 1

If the largest integer drawn among the four cards is 5, then that means that 3 of the cards drawn are from the set {1, 2, 3, 4}. The number of ways of selecting 3 cards from 4 cards is given by:

$^4C_3=4$

If the largest integer drawn among the four cards is 6, then that means that 3 of the cards drawn are from the set {1, 2, 3, 4, 5}. The number of ways of selecting 3 cards from 5 cards is given by:

$^5C_3=10$

If the largest integer drawn among the four cards is 7, then that means that 3 of the cards drawn are from the set {1, 2, 3, 4, 5, 6}. The number of ways of selecting 3 cards from 6 cards is given by:

$^6C_3=20$

Therefore, the number of selections grace could make if the largest integer drawn among the four cards is either a 5, a 6 or a 7 is given by 4 + 10 + 20 = 34 selections.