Problem Solving

Hi. I barely missed this problem as I couldn't recall the formula.. I guess it falls under "Number Properties" Can you help me recall?

12C2 ==> 12x11/2 ==> 66

yumi2012 wrote:Hi. I barely missed this problem as I couldn't recall the formula.. I guess it falls under "Number Properties" Can you help me recall?

Number of ways to select r items out of n unique items = nCr = n! / ((n-r)!*r!)

In this case:
r = 2
n = 12
12C2 = 12! / ((12-2)!*2!)
= 12! / (10! * 2!)
= (12*11)/2
= 6*11
= 66

Choose D

Cheers

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.