Question

If Ali selects 3 cards from 5 different cards. In how many different ways can he select the three cards

Answer 1

Ali is picking cards, by nothing is said about an order, so we assume the order doesn't matter.

To calculate the number of ways to select the 3 cards, we can think is steps.

First, we have 5 possibilities to pick from, so we write a 5:

$5$

Then, we have one less option, that is, 4 options, so we multiply by 4:

$5\cdot4$

For the last card, we have one less option, so just 3, thus we multiply by 3:

$5\cdot4\cdot3$

However, this include the various different orders, but we already assumed the order doesn't matter, so we need to divide by the number of way we can permute these 3 cards, which is 3 times 2 times 1. So we can write like that:

$\frac{5\cdot4\cdot3}{3\cdot2\cdot1}$

Before we calculate all, we can cancel the threes and the 4 with the 2:

$\frac{5\cdot2\cdot1}{1\cdot1\cdot1}=5\cdot2=10$

Thus, there are 10 ways to select the cards.