Question

In how many ways can we select 5 books out of 10, of which a particular book is not included ?

• A. 108
• B. 120
• C. 124
• D. 126

Answer 1

Answer:

D. 126

To find the number of ways to select the books, we use the combination formula:

$^nC_r = \frac{n!}{r!(n-r)!}$

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1. Analyze the conditions:

   • Total number of books available = 10

   • Number of books to select = 5

   • Condition: 1 specific book must not be included.

2. Adjust the pool of choices:

   • Since that particular book cannot be chosen, we remove it completely from our choices.

   • New total books available ($n$) = $10 - 1 = \mathbf{9}$

   • Number of books we still need to select ($r$) = 5

3. Calculate the combinations ($^9C_5$):
   $^9C_5 = \frac{9 \times 8 \times 7 \times 6 \times 5}{5 \times 4 \times 3 \times 2 \times 1}$

4. Simplify the calculation:

   • Cancel out the $5$ from the top and bottom.

   • $4 \times 2 = 8$, which cancels out the $8$ on top.

   • Divide the $6$ on top by the $3$ on the bottom to get $2$.

   • This leaves: $9 \times 7 \times 2 = \mathbf{126}$

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Final Answer

There are 126 ways to select the books under the given conditions.