Find the value of `8C_3`.

Options

A. 56
B. `8!`
C. 65
D. `3^8`

Answer

To find the value of \( 8C_3 \), we will use the formula for combinations, which is given by:

\[ nCr = \frac{n!}{r!(n-r)!} \]

### Step 1: Identify the values of \( n \) and \( r \)
In this case, \( n = 8 \) and \( r = 3 \).

### Step 2: Substitute the values into the formula
We can substitute \( n \) and \( r \) into the formula:

\[ 8C_3 = \frac{8!}{3!(8-3)!} = \frac{8!}{3! \cdot 5!} \]

### Step 3: Expand the factorials
Now we need to expand the factorials. We can write \( 8! \) as \( 8 \times 7 \times 6 \times 5! \):

\[ 8C_3 = \frac{8 \times 7 \times 6 \times 5!}{3! \cdot 5!} \]

### Step 4: Cancel out \( 5! \)
The \( 5! \) in the numerator and the denominator cancels out:

\[ 8C_3 = \frac{8 \times 7 \times 6}{3!} \]

### Step 5: Calculate \( 3! \)
Now, we calculate \( 3! \):

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

### Step 6: Substitute \( 3! \) back into the equation
Now we substitute \( 3! \) back into the equation:

\[ 8C_3 = \frac{8 \times 7 \times 6}{6} \]

### Step 7: Simplify the expression
Now we can simplify the expression:

\[ 8C_3 = 8 \times 7 = 56 \]

### Final Answer
Thus, the value of \( 8C_3 \) is \( 56 \).

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