Question

mdm4u1
3.1 organized counting
example 5:
in a deck of cards, there are 52 cards, and 4 suits. how many ways are there to select:

a) a card which is either a red face card or a black ace?

b) a red face and a black ace one after the other?

c) a heart or a queen?

Explanation:

Part (a)

Step1: Count red face cards

Red suits: hearts, diamonds (2 suits). Face cards: Jack, Queen, King (3 per suit). So red face cards: \(2\times3 = 6\).

Step2: Count black aces

Black suits: spades, clubs (2 suits). Aces: 1 per suit. So black aces: \(2\times1 = 2\).

Step3: Apply addition principle (mutually exclusive)

Total ways = red face cards + black aces = \(6 + 2 = 8\).

Step1: Count red face cards (as before)

Red face cards: \(6\) (from part a).

Step2: Count black aces (as before)

Black aces: \(2\) (from part a).

Step3: Apply multiplication principle (sequence)

Total ways = red face cards \(\times\) black aces = \(6\times2 = 12\).

Step1: Count hearts

Hearts: 13 cards (1 suit of 13).

Step2: Count queens

Queens: 4 cards (1 per suit).

Step3: Count heart queens (overlap)

Heart queen: 1 card (queen of hearts).

Step4: Apply inclusion - exclusion principle

Total ways = hearts + queens - heart queens = \(13 + 4 - 1 = 16\).

Answer:

\(8\)

Part (b)