Question

A deck of cards consists of 4 suits (clubs, spades, diamonds, hearts), each suit consisting of 13 values (ace, 2, 3, 4, 5, 6, 7, 8, 9, jack, queen, king). Four people are playing a game of cards and they are each dealt 13 cards randomly. We say that each person is dealt a hand of 13 cards. A suit distribution for a particular hand is a set of four integers, adding up to 13. How many possible hands are there with a suit distribution of {4, 2, 3, 4}?Show all work and write down the final expression (with explanations) whose value gives the correct answer. You do not have to calculate it numerically.

Answer 1

distributioon : { no. of clubs, no. of spades, no. of diamonds, no. of hearts }

for distribution : {4, 2, 3, 4}

it means : { no. of clubs=4, no. of spades=2, no. of diamonds=3, no. of hearts=4 }

no. of ways = (no. of ways to choose 4 clubs)*(no. of ways to choose 2 spades)*(no. of ways to choose 3 diamonds)*(no. of ways to choose 4 hearts)

= 13C4 * 13C2 * 13C3 * 13C4

= 715 * 78 * 286 * 715

= 11404407300

final expression = 13C4 * 13C2 * 13C3 * 13C4

P.S. (please upvote if you find the answer satisfactory)