Question

In how many ways can a pack of 52 cards be divided into 4 sets, three of them having 16 cards each and the fourth just 4 cards?

(a) 16 ! × 52 ! 

(b) \(\frac{52!}{(16!)^3}\)

(c) \(\frac{52!}{(3!)^{16}}\)

(d) \(\frac{52!}{(16!)^3\times(3!)}\)

Answer 1

(d) \(\frac{52!}{(16!)^3\times(3!)}\)

First player can get 16 cards in ⁵²C₁₆ ways 

Second player can get 16 cards in ³⁶C₁₆ ways 

Third player can get 16 cards in ²⁰C₁₆ ways 

Fourth player can get 4 cards in ⁴C₄ ways 

But the first three sets can be interchanged in 3! ways 

∴ Required number of ways = ⁵²C₁₆ × ³⁶C₁₆ × ²⁰C₁₆ × ⁴C₄ × \(\frac1{3!}\)

= \(\frac{52!}{(16!)^3\times(3!)}\)