Bunuel wrote:
A child throws six differently colored candies up in the air. How many different possible groups of at least one candy are there that she could catch in her mouth?
A. 50
B. 51
C. 62
D. 63
E. 72
Since the order of the candies within a group of candies doesn’t matter, this is a combination problem.
Number of ways the group of candies consists of exactly 1 candy: 6C1 = 6
Number of ways the group of candies consists of exactly 2 candies: 6C2 = (6 x 5)/2 = 15
Number of ways the group of candies consists of exactly 3 candies: 6C3 = (6 x 5 x 4)/(3 x 2) = 20
Number of ways the group of candies consists of exactly 4 candies: 6C4 = 6C2 = 15
Number of ways the group of candies consists of exactly 5 candies: 6C5 = 6C1 = 6
Number of ways the group of candies consists of all 6 candies: 6C6 = 1
Therefore the number of ways the group of candies consists of at least 1 candy is 6 + 15 + 20 + 15 + 6 + 1 = 63.
Alternative solution:
This is an “at least one” problem. Therefore, the number of ways the group of candies consists of at least 1 candy is the number of ways the group of candies consists of any number (from 0 to 6 inclusive) of candies, minus the number of ways the group of candies consists of exactly 0 candies.
The number of ways the group of candies consists of any number of candies is 2^6 = 64 and the number of ways the group of candies consists of exactly 0 candies is 6C0 = 1. Thus, the number of ways the group of candies consists of at least 1 candy is 64 - 1 = 63.
Answer: D
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