What is the total number of possible hands of the hand contains exactly one heart?

Suppose we have 24 certain number and the sum is divided by 6, the result is 1 more than 13 of the original number. possible hands with only one heart

We have to have one heart, so we choose 1 heart from the 13 (13 choose 1) in C (13,1) = 13 ways. We choose the remaining 3 cards from the 39 remaining cards in C (39,3) ways. C (39,3) = 39*38*37 / (1*2*3) = 9139 ways

So the number of possible hands is the product of 13 and 9139 = 118807