Question

From a deck of 52 cards, four cards are drawn simultaneously, find the chance that they will be the four honours of the same suit.

Answer 1

Four cards can be drawn from a pack of 52 cards in ⁵²C₄ ways.
i.e. n(S) = ⁵²C₄
Of these, there are four ways to draw all the four honours of a suit, i.e. J, Q, K and A of Spades, Hearts, Diamonds or Clubs.
Let E be the favourable event of that all the four cards drawn are honour cards from the same suit.
Favourable number of events, n(E) = ⁴C₄ or ⁴C₄ or ⁴C₄ or ⁴C₄ = ⁴C₄ + ⁴C₄ + ⁴C₄ + ⁴C₄
Hence, required probability = $\frac{n\left(E\right)}{n\left(S\right)}=\frac{{}^{4}C_4+{}^{4}C_4+{}^{4}C_4+{}^{4}C_4}{{}^{52}C_4}$
$$\begin{gathered} =\frac{4}{{\displaystyle \frac{52\times 51\times 50\times 49}{4\times 3\times 2\times 1}}} \\ =\frac{4}{13\times 17\times 25\times 49} \\ =\frac{4}{270725} \end{gathered}$$