If a team of four persons is to be selected from 8 males and 8 females, then in how many ways can the selections be made to include at least one male.

Options

A. 1550
B. 1675
C. 1725
D. 1750

Answer

To solve the problem of selecting a team of four persons from 8 males and 8 females, ensuring that at least one male is included, we can follow these steps:

### Step 1: Calculate the total number of ways to select 4 persons from 16 (8 males + 8 females).
The total number of ways to choose 4 persons from 16 can be calculated using the combination formula:
\[ \text{Total ways} = \binom{16}{4} \]

Using the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
we have:
\[ \binom{16}{4} = \frac{16!}{4!(16-4)!} = \frac{16!}{4! \cdot 12!} \]

Calculating this gives:
\[ \binom{16}{4} = \frac{16 \times 15 \times 14 \times 13}{4 \times 3 \times 2 \times 1} = 1820 \]

### Step 2: Calculate the number of ways to select 4 females (no males).
Next, we need to find the number of ways to select 4 females from the 8 females:
\[ \text{Ways to select 4 females} = \binom{8}{4} \]

Using the combination formula:
\[ \binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{8!}{4! \cdot 4!} \]

Calculating this gives:
\[ \binom{8}{4} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70 \]

### Step 3: Subtract the number of all-female selections from the total selections.
To find the number of ways to select a team that includes at least one male, we subtract the number of all-female selections from the total selections:
\[ \text{Ways with at least one male} = \text{Total ways} - \text{Ways to select 4 females} \]
\[ \text{Ways with at least one male} = 1820 - 70 = 1750 \]

### Final Answer:
The number of ways to select a team of four persons that includes at least one male is **1750**.

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