Question

Using a standard deck of 52 playing cards:

1. How many different ways can I pull 3 cards out of a deck (without replacement).

2. In how many of those situations in part (1) did I pull a 3 first?
3. In how many of those situations in part (1) did I pull a 3 or 4 first?
4. In how many of those situations in part (1) did I pull a spade or an ace?

Answer 1

Given there are 52 cards

• Number of ways of pulling 3 cards WITHOUT REPLACEMENT = 52C3 = 22100

• In a standard deck, there are 4 cards (Spade, Clubs, Hearts, Diamond) of number 3

Number of ways of pulling a 3 – card first = 4C1 = 4

And then, there would be (52 – 1) = 51 cards to pull the remaining two cards.

Number of ways of pulling two cards from 51 cards = 51C2 = 1275

Therefore, total number of situations in pulling a 3 – card first = 4 x 1275 = 5100

• In a standard deck,

there are 4 cards (Spade, Clubs, Hearts, Diamond) of number 3 and also there are 4 cards (Spade, Clubs, Hearts, Diamond) of number 4

Number of ways of pulling a 3 – card or a 4 – card first = 8C1 = 8

And then, there would be (52 – 1) = 51 cards to pull the remaining two cards.

Number of ways of pulling two cards from 51 cards = 51C2 = 1275

Therefore, total number of situations in pulling a 3 – card or a 4 – card first = 8 x 1275 = 10200

• In a standard deck, there are 13 – cards of Spade (Ace, 1, 2, …. King, Queen Jack) and remaining there would be 3 – Aces (1 each of Hearts, Clubs and Diamonds)

So, total there are 16 cards of Spade and Ace combined

In three cards there can be a total of (1 card of Spade or Ace) or, (2 cards of Spade or Ace) or (3 cards of Spade or Ace)

Number of situations where there can be 1 card of spade or ace = 16C1 x 36C2 = 10080

(36C2 – Since, there should only be 1 card of Spade or Ace, the remaining two cards must be pulled from (52 – 16) = 36 cards)

Simialrly, Number of situations where there can be 2 cards of spade or ace = 16C2 x 36C1 = 4320

And, Number of situations where there can be 3 cards of Spade or Ace = 16C3 = 560

Therefore,

Total number of situations where there can be a spade or ace = 10080 + 4320 + 560 = 14960