If a quality control check is made inspecting a sample of 2 light bulbs from a box of 12 lighbulbs, how many different samples can be chosen?
A) 6
B) 24
C) 36
D) 66
E) 72
Approach 1:
Number of options for the 1st bulb = 12. (Any of the 12 bulbs.)
Number of options for the 2nd bulb = 11. (Any of the 11 remaining bulbs.)
To combine these options, we multiply:
12*11.
Since the ORDER of the bulbs doesn't matter -- AB are the same two bulbs as BA -- we divide by the number of ways to ARRANGE the 2 bulbs (2!):
(12*11)/(2*1) = 66.
The correct answer is
D.
Approach 2:
Let the 12 bulbs be A-B-C-D-E-F-G-H-I-J-K-L.
Bulb A can be paired with any of the following:
B-C-D-E-F-G-H-I-J-K-L.
Total options = 11.
All of the pairs that include Bulb A have been counted.
If we exclude Bulb A, Bulb B can be paired with any of the following:
C-D-E-F-G-H-I-J-K-L.
Total options = 10.
All of the pairs that include Bulbs A and/or B have been counted.
If we exclude Bulbs A and B, Bulb C can be paired with any of the following:
D-E-F-G-H-I-J-K-L.
Total options = 9.
Note the PATTERN.
The results are descending consecutive integers, starting with 11.
Implication:
The total number of pairs is equal to the sum of the integers between 11 and 1, inclusive.
Thus:
Total possible pairs = 11+10+9+8+7+6+5+4+3+2+1 = 66.
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