Four cards are drawn at random from a pack of 52 playing cards. Find the probability of getting four honours of the same suit.

Answer

To find the probability of drawing four honors of the same suit from a standard deck of 52 playing cards, we can follow these steps:

### Step 1: Understand the Problem
We need to find the probability of drawing 4 cards that are all honors (Jack, Queen, King, Ace) and all from the same suit.

### Step 2: Identify Total Number of Ways to Choose 4 Cards
The total number of ways to choose 4 cards from a deck of 52 cards is given by the combination formula:
\[ \text{Total ways} = \binom{52}{4} \]
Calculating this:
\[ \binom{52}{4} = \frac{52!}{4!(52-4)!} = \frac{52 \times 51 \times 50 \times 49}{4 \times 3 \times 2 \times 1} = 270725 \]

### Step 3: Identify the Number of Favorable Outcomes
Each suit (hearts, diamonds, clubs, spades) has 4 honors: Jack, Queen, King, and Ace. We need to select all 4 honors from one suit.

1. The number of ways to choose 4 honors from 4 available honors in one suit is:
\[ \binom{4}{4} = 1 \]
2. Since there are 4 suits, the total number of favorable outcomes is:
\[ \text{Favorable outcomes} = 4 \times \binom{4}{4} = 4 \times 1 = 4 \]

### Step 4: Calculate the Probability
The probability \( P \) of drawing 4 honors of the same suit is given by the ratio of the number of favorable outcomes to the total number of outcomes:
\[ P = \frac{\text{Favorable outcomes}}{\text{Total ways}} = \frac{4}{270725} \]

### Step 5: Simplify the Probability
This fraction can be simplified if necessary, but it is already in its simplest form. Therefore, the final probability is:
\[ P = \frac{4}{270725} \]

### Final Answer
The probability of drawing four honors of the same suit from a standard deck of 52 playing cards is:
\[ \frac{4}{270725} \]