We wish to choose 7 cards from a usual deck of 52 playing cards. In how many ways can this be done if we are required to choose the cards in the following ways?

(a) with no restriction.

(b) all cards come from the same suit.

(c) exactly 3 Aces and exactly 3 Kings are chosen.

(d) all 7 cards have values between 2 and 7 inclusive.

(e) all 7 cards all have different values (where Jacks are different from Queens, etc.).

In a usual deck of 52 cards, there are 4 suits with 13 card for each suit.

(a) with no restriction.

If there is no restriction and different order is not important, then the possible way should be: 52! / (52-7) !7! = 52!/45!7! = (52 * 51 * * 50*49*48*47*46) / (7*6*5*4*3*2*1) = 674274182400/5040 = 133,784,560 ways

(b) all cards come from the same suit.

If all card has to come from the same suit, that means you will only have 13 possible cards for each suit.

Then the possible ways for each suit: 13! / (13-7) !7! = (13*12*11*10*9*8*7) / (7*6*5*4*3*2) = 8648640/5040 = 1716 ways per suit.

Since there are 4 suits then the possible way = 1716 * 4 = 6864

(c) exactly 3 Aces and exactly 3 Kings are chosen.

In this case, we will have 3 aces, 3 kings, and one random card. There are 4 aces/kings in one deck and we have to choose 4.

The possible ways for 3 aces or 3 kings would be: 4!/3! (4-3) ! = 4 ways.

After choosing the 3 aces and 3 kings, the deck should have 52-3-3 = 46 card left. That mean there will be 46 possible ways for the random card.

The total possibilities should be: 4 * 4 * 46 = 736 ways

(d) all 7 cards have values between 2 and 7 inclusive

There are four cards with each value. The value between 2 and 7 inclusive should be 7-2+1 = 6 different value. Since each value has 4 cards and we have 6 different value, then the total possible card is 6*4 = 24

Then, we take 7 cards from those 24 cards. The possible should be: 24!/7! (24-7) ! = (24*23*22*21*20*19*18) / 5040 = 1744364160/5040 = 346104

(e) all 7 cards all have different values (where Jacks are different from Queens, etc.).

There is 4 card with same value. For the first take, you will have 52 different ways. But for the second take, you will only have 52-4=48 different ways. That was because you can use:

1. one card that you take earlier.

2. three cards with the same value.

Then the possible ways become like this: 52*48*44*40*36*32*28 / 7! = 28,114,944