Evaluate the expression. \[ \frac{{ }_{8} C_{2} \cdot{ }_{8} C_{1}}{{ }_{24} C_{3}} \] \( \frac{{ }_{8} C_{2} \cdot{ }_{8} C_{1}}{{ }_{24} C_{3}}=\square \) (Type an integer or a simplified fraction.)

To solve the expression \( \frac{{ }_{8} C_{2} \cdot{ }_{8} C_{1}}{{ }_{24} C_{3}} \), we first need to calculate each of the binomial coefficients involved. Binomial coefficients are given by the formula: \[ _{n}C_{k} = \frac{n!}{k!(n-k)!} \] 1. Calculate \( _{8}C_{2} \): \[ _{8}C_{2} = \frac{8!}{2!(8-2)!} = \frac{8!}{2! \cdot 6!} = \frac{8 \cdot 7}{2 \cdot 1} = \frac{56}{2} = 28 \] 2. Calculate \( _{8}C_{1} \): \[ _{8}C_{1} = \frac{8!}{1!(8-1)!} = \frac{8!}{1! \cdot 7!} = 8 \] 3. Calculate \( _{24}C_{3} \): \[ _{24}C_{3} = \frac{24!}{3!(24-3)!} = \frac{24!}{3! \cdot 21!} = \frac{24 \cdot 23 \cdot 22}{3 \cdot 2 \cdot 1} \] Calculating the numerator: \[ 24 \cdot 23 = 552 \] \[ 552 \cdot 22 = 12144 \] Now, for the denominator: \[ 3! = 6 \] So, \[ _{24}C_{3} = \frac{12144}{6} = 2024 \] 4. Now substituting these calculations back into the expression: \[ \frac{_{8}C_{2} \cdot _{8}C_{1}}{_{24}C_{3}} = \frac{28 \cdot 8}{2024} \] Calculate the numerator: \[ 28 \cdot 8 = 224 \] Now, we substitute this into the expression: \[ \frac{224}{2024} \] To simplify this fraction, we can find the greatest common divisor (GCD) of 224 and 2024. Calculating the GCD: \[ 224 = 2^5 \cdot 7 \] \[ 2024 = 2^2 \cdot 3 \cdot 13 \cdot 7 \] The GCD is \( 2^2 \cdot 7 = 28 \). Now we divide the numerator and the denominator by their GCD: \[ \frac{224 \div 28}{2024 \div 28} = \frac{8}{72} = \frac{1}{9} \] Thus, the final answer is: \[ \frac{1}{9} \] The evaluated expression is: \[ \boxed{\frac{1}{9}} \]