Question 39 of 42intermediate🔧 ApplyNumerical3 marks

Determine the number of 5-card combinations out of a deck of 52 cards if each selection of 5 cards has exactly one king.

Correct Answer

778320

Exercise: Miscellaneous Exercise on Chapter 6 | Q: 8 | (Page 24)
For More Understanding

Explanation

This question is similar to Question 6 in the Miscellaneous Exercise which asks for exactly one ace. The same logic applies: we use the multiplication principle by first selecting 1 king from 4 kings, then selecting 4 non-king cards from the remaining 48 cards. The formula for combinations nCr is used as shown in Example 19.

Solution Steps

  1. Step 1: A deck has 52 cards with 4 kings. We need exactly one king in our selection of 5 cards.

  2. Step 2: Number of ways to select 1 king from 4 kings = ⁴C₁ = 4!/(1! × 3!) = 4

  3. Step 3: Number of ways to select remaining 4 cards from 48 non-king cards = ⁴⁸C₄ = 48!/(4! × 44!)

  4. Step 4: Calculating ⁴⁸C₄ = (48×4748 \times 47 × 46×4546 \times 45)/(4×34 \times 3 × 2×12 \times 1) = 194580

  5. Step 5: By multiplication principle, total combinations = ⁴C₁ × ⁴⁸C₄ = 4×1945804 \times 194580 = 778320

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