In how many ways can a pack of 52 cards be divided into 4 sets, three

1.	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» (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 × \(\frac1{3!}\) = \(\frac{52!}{(16!)^3\times(3!)}\)

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»

(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 × \(\frac1{3!}\)

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