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

Solution:

The total no. of cards in a deck of cards = 52.

Among these, no. of aces = 4.

We know that the number of ways of selecting r different things from n different things is a combination and is calculated using the formula ⁿCᵣ = n! / [r!(n−r)!].

Exactly 1 ace out of 4 aces can be selected in ⁴C₁ ways.

Total number of cards to be selected = 5 (among which 1 (ace) is already selected).

There are 52 - 4 = 48 non-aces to select the remaining 4 cards.

No. of ways of selecting 4 cards from the remaining deck of 48 cards = ⁴⁸C₄

By fundamental principle of counting,

The required number of ways = ⁴C₁ × ⁴⁸C₄

= (4!) / [1! (4-1)!] × (48!) / [4! (48-4)!]  (Using nCr formula)

= 778320

NCERT Solutions Class 11 Maths Chapter 7 Exercise 7.4 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.

Summary:

The number of 5 card combinations out of a deck of 52 cards if there is exactly one ace in each combination is 778320