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Evaluate the expression:

[tex]\[ \frac{{ }_8 C_3}{{ }_4 C_3} - \frac{811}{791} \][/tex]

[tex]\[ \frac{{ }_8 C_3}{{ }_4 C_3} - \frac{811}{791} = \square \][/tex]

(Type an integer or a simplified fraction.)

Sagot :

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]