5 books are to be chosen from a lot of 10 books.If m is the number of ways of choice when one specified book is always included and n is the number of ways of choice when a specified book is always excluded, then which one of the following is correct?

Options

A. `m gt n`
B. `m=n`
C. m=n-1
D. m=n-2

Answer

To solve the problem, we need to calculate the number of ways to choose books under two different conditions: when a specified book is always included and when a specified book is always excluded.

### Step-by-step Solution:

1. **Total Books**: We have a total of 10 books.

2. **Condition 1 (One specified book is always included)**:
- If one specified book is always included, we can consider that book as already chosen.
- This leaves us with 9 books (10 - 1 = 9) from which we need to choose 4 more books (since we need a total of 5 books).
- The number of ways to choose 4 books from 9 is given by the combination formula \( C(n, r) \), which is calculated as:
\[ m = \binom{9}{4} = \frac{9!}{4! \cdot (9-4)!} = \frac{9!}{4! \cdot 5!} \]

3. **Condition 2 (One specified book is always excluded)**:
- If one specified book is always excluded, we are left with 9 books (since we exclude that one book).
- In this case, we need to choose 5 books from these 9 available books.
- The number of ways to choose 5 books from 9 is given by:
\[ n = \binom{9}{5} = \frac{9!}{5! \cdot (9-5)!} = \frac{9!}{5! \cdot 4!} \]

4. **Comparing m and n**:
- We have calculated:
\[ m = \binom{9}{4} \quad \text{and} \quad n = \binom{9}{5} \]
- By the property of combinations, we know that:
\[ \binom{n}{r} = \binom{n}{n-r} \]
- Therefore:
\[ \binom{9}{4} = \binom{9}{5} \]
- This implies that:
\[ m = n \]

### Conclusion:
The relationship between \( m \) and \( n \) is:
\[ m = n \]

### Final Answer:
The correct option is **m is equal to n**.

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