Question

Question 6 Determine the number of 5 card combinations out of a deck of 52 cards if there is exactly one ace in each combination.

Class X1 - Maths -Permutations and Combinations Page 153

Answer 1

in a pack of 52 cards , there are four aces.
I.e one ace will be selected from 4 cards and remaining 4 cards will be selected from rest 48 cards .

number of ways selecting one ace from 4 aces = ⁴C₁
number of ways selecting 4 cards from 48 cards = ⁴⁸C₄

now, A/C to concept of fundamental principle of counting,
5 cards with exactly one ace can be selected in ⁴C₁ × ⁴⁸C₄ ways .

= 4!/3! × 48!/44!×4!
= 48 × 47 × 46 × 45/6
= 8 × 47 × 46 × 45
= 778320 ways .

Answer 2

778320 ways

Step-by-step explanation:

Given:

Total number of cards = 52

Number of ace cards = 4

Number of non ace cards = 52 - 4 = 48

Solving:

One ace card out of 4 can be selected in 4C1 ways.

Remaining 4 cards out of 48 cards can be selected in 48 C4 ways.

∴ Required no. of ways of selecting 5 cards:

= 4C1 * 48C4 = $\frac{4!}{3! x 1!} *\frac{48!}{4!*44!}$

= $\frac{48 * 47*46*45}{6}$ = 778320

Therefore there are 778320 ways.