Find the value of :
` ""^(10)C_4`

Answer

To find the value of \( \binom{10}{4} \), we will use the formula for combinations:

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

In this case, \( n = 10 \) and \( r = 4 \). Therefore, we can substitute these values into the formula:

\[ \binom{10}{4} = \frac{10!}{4!(10-4)!} = \frac{10!}{4! \cdot 6!} \]

Next, we can express \( 10! \) in a way that allows us to cancel \( 6! \):

\[ 10! = 10 \times 9 \times 8 \times 7 \times 6! \]

Now substituting this back into our equation gives:

\[ \binom{10}{4} = \frac{10 \times 9 \times 8 \times 7 \times 6!}{4! \cdot 6!} \]

We can cancel \( 6! \) from the numerator and the denominator:

\[ \binom{10}{4} = \frac{10 \times 9 \times 8 \times 7}{4!} \]

Next, we need to calculate \( 4! \):

\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

Now we can substitute this value back into our equation:

\[ \binom{10}{4} = \frac{10 \times 9 \times 8 \times 7}{24} \]

Calculating the numerator:

\[ 10 \times 9 = 90 \]
\[ 90 \times 8 = 720 \]
\[ 720 \times 7 = 5040 \]

Now we have:

\[ \binom{10}{4} = \frac{5040}{24} \]

Finally, we perform the division:

\[ \frac{5040}{24} = 210 \]

Thus, the value of \( \binom{10}{4} \) is:

\[ \boxed{210} \]