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:

Step1: Identify number of red face - cards and black aces

There are 6 red face - cards (3 face - cards in hearts and 3 in diamonds) and 2 black aces (ace of spades and ace of clubs).

Step2: Use the addition principle for non - overlapping events (part a)

For part a, since a card cannot be both a red face - card and a black ace at the same time, the number of ways to select a card which is either a red face - card or a black ace is \(6 + 2=8\).

Step3: Use the multiplication principle for sequential events (part b)

For part b, the number of ways to select a red face - card first and then a black ace is \(6\times2 = 12\), and the number of ways to select a black ace first and then a red face - card is \(2\times6=12\). So the total number of ways to select a red face and a black ace one after the other is \(12 + 12=24\).

Step4: Use the inclusion - exclusion principle (part c)

There are 13 hearts and 4 queens in a deck. But the queen of hearts is counted in both sets. So the number of ways to select a heart or a queen is \(13+4 - 1=16\).

Answer:

a) 8
b) 24
c) 16