If the number of ways in which a lawn-tennis mixed double be made from seven married couples if no husband and wife play in the same set is K , then greatest prime divisor of K is __________.

Answer

To solve the problem of finding the greatest prime divisor of \( K \), where \( K \) is the number of ways to form a mixed doubles team from 7 married couples with the condition that no husband and wife play in the same set, we can follow these steps:

### Step 1: Understand the Composition of the Teams
We have 7 married couples, which means there are 7 men and 7 women. For a mixed doubles team, we need 1 man and 1 woman.

### Step 2: Choose the Men
We need to select 2 men from the 7 available men. The number of ways to choose 2 men from 7 can be calculated using permutations because the order in which we choose the men matters. This is given by:
\[ 7P2 = \frac{7!}{(7-2)!} = \frac{7!}{5!} = 7 \times 6 = 42 \]

### Step 3: Exclude Their Wives
Since no husband and wife can play in the same set, if we have chosen 2 men, we cannot choose their wives. This leaves us with 5 women to choose from.

### Step 4: Choose the Women
Now, we need to select 2 women from the remaining 5 women. The number of ways to choose 2 women from 5 is also calculated using permutations:
\[ 5P2 = \frac{5!}{(5-2)!} = \frac{5!}{3!} = 5 \times 4 = 20 \]

### Step 5: Calculate Total Combinations
The total number of ways to form the mixed doubles team \( K \) is the product of the number of ways to choose the men and the number of ways to choose the women:
\[ K = 7P2 \times 5P2 = 42 \times 20 = 840 \]

### Step 6: Factorize \( K \)
Next, we need to find the prime factorization of \( K \):
\[ 840 = 2^3 \times 3^1 \times 5^1 \times 7^1 \]

### Step 7: Identify the Greatest Prime Divisor
From the prime factorization, we can see the prime factors are 2, 3, 5, and 7. The greatest of these prime factors is:
\[ \text{Greatest Prime Divisor} = 7 \]

### Final Answer
Thus, the greatest prime divisor of \( K \) is \( \boxed{7} \).

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