Question 28 of 42intermediate🔧 ApplyNumerical3 marks

Determine the number of 5 card combinations out of a deck of 52 cards if there is exactly one ace in each combination.

Correct Answer

We know that a deck has 52 cards with 4 suits. Since there is one ace in each suit, there are 4 aces in total. The number of non-ace cards is 52 - 4 = 48. We need to select 5 cards with exactly one ace. This requires selecting 1 ace from the 4 available aces and the remaining 4 cards from the 48 non-ace cards.

Number of ways to select 1 ace:

4C1=4^4C_1 = 4

Number of ways to select 4 non-ace cards:

48C4=48×47×46×454×3×2×1=194580^{48}C_4 = \frac{48 \times 47 \times 46 \times 45}{4 \times 3 \times 2 \times 1} = 194580

Total number of combinations: Using the multiplication principle, the required number of ways is:

4C1×48C4=4×194580=778320^4C_1 \times ^{48}C_4 = 4 \times 194580 = 778320
Exercise: EXERCISE 6.4 | Q: 6 | (Page 20)
For More Understanding

Explanation

The solution follows the logic presented in Example 19 and the Miscellaneous Examples of the provided text. The text establishes there are 4 suits, implying 4 aces. The method uses the combination formula nCr^nC_r to choose items where order is not important, and the multiplication principle to combine the independent choices of selecting aces and non-aces.

Solution Steps

  1. Step 1: Determine the count of aces and non-aces. There are 4 aces (one per suit) and 48 non-ace cards (52 - 4).

  2. Step 2: Calculate ways to choose 1 ace from 4 aces using 4C1^4C_1, which equals 4.

  3. Step 3: Calculate ways to choose 4 non-ace cards from 48 using 48C4^{48}C_4, which equals 194580.

  4. Step 4: Multiply the results using the multiplication principle: 4×194580=7783204 \times 194580 = 778320.

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