Let's evaluate the given expression step-by-step:
[tex]\[
\frac{{ }_8 C_3}{{ }_4 C_3}-\frac{811}{791}
\][/tex]
1. Calculate [tex]\({ }_8 C_3\)[/tex]:
The combination formula is given by:
[tex]\[
{ }_n C_r = \frac{n!}{r!(n-r)!}
\][/tex]
Substituting [tex]\( n = 8 \)[/tex] and [tex]\( r = 3 \)[/tex], we get:
[tex]\[
{ }_8 C_3 = \frac{8!}{3!(8-3)!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56
\][/tex]
2. Calculate [tex]\({ }_4 C_3\)[/tex]:
Substituting [tex]\( n = 4 \)[/tex] and [tex]\( r = 3 \)[/tex], we get:
[tex]\[
{ }_4 C_3 = \frac{4!}{3!(4-3)!} = \frac{4 \times 3 \times 2 \times 1}{3 \times 2 \times 1 \times 1} = 4
\][/tex]
3. Evaluate the fraction [tex]\(\frac{{ }_8 C_3}{{ }_4 C_3}\)[/tex]:
[tex]\[
\frac{56}{4} = 14
\][/tex]
4. Evaluate the fraction [tex]\(\frac{811}{791}\)[/tex]:
The value of this fraction is approximately:
[tex]\[
\frac{811}{791} \approx 1.0253
\][/tex]
5. Subtract the two results:
[tex]\[
14 - 1.0253 \approx 12.9747
\][/tex]
Therefore, the result of the expression
[tex]\[
\frac{{ }_8 C_3}{{ }_4 C_3} - \frac{811}{791}
\][/tex]
is approximately:
[tex]\[
12.9747
\][/tex]
