In a hand of 5 cards how many ways are there to get exactly 3 kings or 4 kings?

Question

In a hand of 5 cards how many ways are there to get exactly 3 kings or 4 kings?

Answer 1

ways to get 3 or 4 kings
= 4C3*48C2 + 4C4*48C1
= 4*1128 + 1*48
= 4560

Answer 2

[Edit -- Linka K is counting "permutations" when she should be counting "combinations." And I have no idea what "osamu y" is counting.]

Imagine you're laying out 5 cards one at a time, in such a way that the first 3 are always kings. (In addition, there may or may not be a king among the last 2 cards.)

For the 1st card, you have 4 choices (any of the 4 kings).
For the 2nd card, you have 3 choices (any of the 3 remaining kings).
For the 3rd card, you have 2 choices (either if the 2 remaining kings).
For the 4th card, you have 49 choices (any of the 49 remaining cards).
For the 5th card, you have 48 choices (any of the 48 remaining cards).

So, the total number of ways in which you can lay out 5 cards in this fashion is: 4*3*2*49*48 = 84672.

However, some of those layouts are the "same" as some of the others. For example, the following are two distinct layouts but they actually represent the same "hand":

Kd; Kh; Ks; 8s; 3c
Kh; Ks; Kd; 3c; 8s

There are actually 12 different layouts (where the first 3 cards are kings) that represent this identical hand. So in order to avoid duplicates, we need to divide the 84672 layouts by 12. The number 12 accounts for the 6 different ways of arranging the three leading kings, and the 2 ways of arranging the last two cards.

But we're not done. In cases where there are 4 kings, there are additional "equivalent" layouts that involve swapping a king in the first 3 with a king in the last 2. For example, these two layouts represent the same hand:

Kh; Kd; Ks; Ah; Kc
Kh; Kc; Ks; Ah; Kd

Any time you have a king in the last two, there are 3 other kings you can swap it with to get another layout that's equivalent. This means you need to divide the total by 3 to get rid of these kinds of duplicates.

So finally, this means the number you want is:

84672 / 12 / 3 = 2352

Answer 3

K K K __ __ or K K K K __
4*3*2*48*47 + 4*3*2*1*48
54144+ 1152
55296 ways

Answer 4

4*50*49/2+1*50=4950

I assume you count in the two jokers, so a whole card set should contain 54 cards.