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 =\[\frac{n\left( E \right)}{n\left( S \right)} = \frac{^{4}{}{C}_4 + ^{4}{}{C}_4 + ^{4}{}{C}_4 + ^{4}{}{C}_4}{^{52}{}{C}_4}\]

\[= \frac{4}{\frac{52 \times 51 \times 50 \times 49}{4 \times 3 \times 2 \times 1}}\]
\[ = \frac{4}{13 \times 17 \times 25 \times 49}\]
\[ = \frac{4}{270725}\]