Find the number of ways, in which 7 books can be selected out of 10 books available.

Answer

To find the number of ways to select 7 books from a total of 10 books, we will use the concept of combinations. The formula for combinations is given by:

\[ C(n, r) = \frac{n!}{(n - r)! \cdot r!} \]

Where:
- \( n \) is the total number of items (in this case, books),
- \( r \) is the number of items to choose.

### Step 1: Identify \( n \) and \( r \)
Here, we have:
- \( n = 10 \) (the total number of books),
- \( r = 7 \) (the number of books we want to select).

### Step 2: Substitute into the combination formula
Now, we substitute \( n \) and \( r \) into the combination formula:

\[ C(10, 7) = \frac{10!}{(10 - 7)! \cdot 7!} \]

### Step 3: Simplify the expression
Calculate \( 10 - 7 = 3 \), so we have:

\[ C(10, 7) = \frac{10!}{3! \cdot 7!} \]

### Step 4: Expand the factorials
Now, we can expand \( 10! \):

\[ 10! = 10 \times 9 \times 8 \times 7! \]

So, we can rewrite our expression:

\[ C(10, 7) = \frac{10 \times 9 \times 8 \times 7!}{3! \cdot 7!} \]

### Step 5: Cancel out \( 7! \)
The \( 7! \) in the numerator and denominator cancels out:

\[ C(10, 7) = \frac{10 \times 9 \times 8}{3!} \]

### Step 6: Calculate \( 3! \)
Now, calculate \( 3! \):

\[ 3! = 3 \times 2 \times 1 = 6 \]

### Step 7: Substitute back into the equation
Now substitute \( 3! \) back into the equation:

\[ C(10, 7) = \frac{10 \times 9 \times 8}{6} \]

### Step 8: Perform the multiplication
Now, calculate the multiplication in the numerator:

\[ 10 \times 9 = 90 \]
\[ 90 \times 8 = 720 \]

### Step 9: Divide by \( 6 \)
Now, divide by \( 6 \):

\[ C(10, 7) = \frac{720}{6} = 120 \]

### Final Answer
Thus, the number of ways to select 7 books out of 10 is:

\[ \boxed{120} \]

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