# A box of candy hearts contains 60 hearts, of which 12 are white, 18 are tan, 9 are pink, 10 are...

## Question:

A box of candy hearts contains 60 hearts, of which 12 are white, 18 are tan, 9 are pink, 10 are yellow, and 11 are green. Suppose you randomly select 2 hearts from the box. Provided the two selected hearts are of the same color, what is the probability that they are neither pink nor yellow?

Step 1 : Compute number of pairs

Number of pairs of white color = {eq}\left ( \begin{array}2 12\\ 2 \end {array} \right ) = \frac{12!}{10! 2!} = 66 {/eq}

Number of pairs of tan color = {eq}\left ( \begin{array}2 18\\ 2 \end {array} \right ) = \frac{18!}{16! 2!} = 153 {/eq}

Number of pairs of pink color = {eq}\left ( \begin{array}2 9\\ 2 \end {array} \right ) = \frac{9!}{7! 2!} = 36 {/eq}

Number of pairs of yellow color = {eq}\left ( \begin{array}2 10\\ 2 \end {array} \right ) = \frac{10!}{8! 2!} = 45 {/eq}

Number of pairs of green color = {eq}\left ( \begin{array}2 11\\ 2 \end {array} \right ) = \frac{11!}{9! 2!} = 55 {/eq}

Total number of pairs is 66 + 153 + 36 +45 + 55= 355

Provided both candies are of the same color, the probability of them not being pink nor yellow is {eq}1 - \frac{36}{355} - \frac{45}{355} = 0.77183 {/eq}