Three cards are drawn at random, one after the other, without replacement from a well-shuffled pack of 52 playing cards. The probability that the three cards are of three different suits is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{169}{425}\)
The first card can be of any suit. After it, 51 cards are left, and 39 of them are of the other three suits. After two cards of different suits, 50 cards are left, and 26 of them are of the remaining two suits. By the multiplication theorem:
\( \displaystyle P = 1\times\frac{39}{51}\times\frac{26}{50} \)
\( \displaystyle = \frac{13}{17}\times\frac{13}{25} = \frac{169}{425} \)
Why the other options are incorrect
Option (2) \(\displaystyle\frac{3}{8}\) — Incorrect. This is \(1\times\frac{3}{4}\times\frac{2}{4}\), the answer if each card were put back before the next draw.
Option (3) \(\displaystyle\frac{169}{2550}\) — Incorrect. This counts only the 4 choices of three suits and forgets their order; the three suits can come in \(4\times 3\times 2 = 24\) orders, each with probability \(\frac{13\times 13\times 13}{52\times 51\times 50}\).
Option (4) \(\displaystyle\frac{13}{17}\) — Incorrect. This is only the probability that the second card differs in suit from the first.