In how many ways can a committee of 8 be chosen from 10 individuals?

Answer

To solve the problem of how many ways a committee of 8 can be chosen from 10 individuals, we can use the concept of combinations. Here’s a step-by-step solution:

### Step 1: Identify the combination formula
The number of ways to choose \( r \) objects from \( n \) objects is given by the combination formula:
\[ nCr = \frac{n!}{r!(n-r)!} \]

### Step 2: Apply the formula to the problem
In this case, we need to choose 8 individuals from a total of 10. So, we need to calculate \( 10C8 \):
\[ 10C8 = \frac{10!}{8!(10-8)!} = \frac{10!}{8! \cdot 2!} \]

### Step 3: Simplify the factorials
We can simplify \( 10! \):
\[ 10! = 10 \times 9 \times 8! \]
Now substituting this back into the equation:
\[ 10C8 = \frac{10 \times 9 \times 8!}{8! \cdot 2!} \]

### Step 4: Cancel out the common terms
The \( 8! \) in the numerator and denominator cancels out:
\[ 10C8 = \frac{10 \times 9}{2!} \]

### Step 5: Calculate \( 2! \)
Now calculate \( 2! \):
\[ 2! = 2 \times 1 = 2 \]

### Step 6: Substitute and calculate
Now substitute \( 2! \) back into the equation:
\[ 10C8 = \frac{10 \times 9}{2} = \frac{90}{2} = 45 \]

### Final Answer
Thus, the number of ways to choose a committee of 8 from 10 individuals is:
\[ \boxed{45} \]