Hello, shaehl!

Surely, you're familiar with Combinations and Factorials ?!

The problems on this page all deal with hands taken from a deck of 32 cards:

the 7, 8, ... , Q, K, A of each of the four suits.

1) How many five-card hands are possible which satisfy:

. . (a) the hand has exactly two Hearts

. . (b) the hand has exactly two Spades

. . (c) the hand has exactly two Kings(a) Two Hearts

There are 8 Hearts and 24 Others.

We want 2 Hearts and 3 Others.

To get 2 Hearts, there are: . ways.

To get 3 Others, there are: . ways.

Therefore: .

(b) Two Spades

The reasoning (and the answer) is identical to part (a).

(c) Two Kings

There are 4 Kings and 28 Others.

We want 2 Kings and 3 Others.

To get 2 Kings, there are: . ways.

To get 3 Others, there are: . ways.

Therefore: .

2) How many 6-card hands are possible which satisfy:

. . (a) the hand has exactly two Hearts

. . (b) the hand has exactly three Spades

. . (c) the hand has exactly 2 Kings(a) Two Hearts

There are 8 Hearts and 24 Others.

We want 2 Hearts and 4 Others.

To get 2 Hearts, there are: . ways.

To get 4 Others, there are: . ways.

Therefore: .

(b) Three Spades

There are 8 Spades and 24 Others.

We want 3 Spades and 3 Others.

To get 3 Spades, there are: . ways.

To get 3 Others, there are: . ways.

Therefore: .

(c) Two Kings

There are 4 Kings and 28 Others.

We want 2 Kings and 4 Others.

To get 3 Kings, there are: . ways.

To get 4 Others, there are: . ways.

Therefore: .

Is there a typo?3) How many 6-card hands are possible which satisfy:

. . (a) the hand has exactly two Hearts

. . (b) the hand has exactly three Spades

This is identical to problem #2.