Q8:

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

Step 1: Analyzing the Selection Criteria

The problem requires the selection of a 5-card combination from a standard deck of 52 cards, subject to the constraint that exactly one card must be a King. A standard deck contains:

• Total cards: $52$
• Total Kings: $4$
• Total Non-Kings: $48$

[Per the Fundamental Principle of Counting and the definition of Combinations, the total selection is the product of independent selections from mutually exclusive subsets.]

Step 2: Formulating the Combination Components

To satisfy the requirement of exactly one King, the selection must be split into two distinct parts:

1. Selection of the King: We must choose $1$ King from the $4$ available Kings. This is denoted as ${^4C_1}$.
2. Selection of the remaining cards: Since a 5-card hand is required and $1$ card is already a King, $4$ additional cards must be chosen from the remaining non-king cards ($48$). This is denoted as ${^{48}C_4}$.

Step 3: Algebraic Calculation

The total number of combinations is given by the product: ${^4C_1} \times {^{48}C_4}$.

Using the combination formula ${^nC_r} = \frac{n!}{r!(n-r)!}$:

${^4C_1} = \frac{4!}{1!(4-1)!} = \frac{4}{1} = 4$

${^{48}C_4} = \frac{48!}{4!(48-4)!} = \frac{48 \times 47 \times 46 \times 45}{4 \times 3 \times 2 \times 1}$

Step 4: Simplifying the Calculation

${^{48}C_4} = \frac{48}{24} \times 47 \times 46 \times 45 = 2 \times 47 \times 46 \times 45$

${^{48}C_4} = 194,580$

Now, multiply by the selection of the King:

Total Combinations $= 4 \times 194,580 = 778,320$

Visual Representation of Selection Logic

Final Solution: 778,320