Questions: Evaluate the expression. (4 C2 * 7 C1) / (21 C3) = □ (Type an integer or a simplified fraction.)

Solution Steps

The combination formula is given by: \[ {}_{n}C_{k} = \frac{n!}{k!(n-k)!} \] For \( {}_{4}C_{2} \): \[ {}_{4}C_{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3 \times 2!}{2! \times 2!} = \frac{12}{2} = 6 \]

For \( {}_{7}C_{1} \): \[ {}_{7}C_{1} = \frac{7!}{1!(7-1)!} = \frac{7 \times 6!}{1 \times 6!} = 7 \]

For \( {}_{21}C_{3} \): \[ {}_{21}C_{3} = \frac{21!}{3!(21-3)!} = \frac{21 \times 20 \times 19 \times 18!}{6 \times 18!} = \frac{7980}{6} = 1330 \]

Now, substitute the values into the original expression: \[ \frac{{}_{4}C_{2} \cdot {}_{7}C_{1}}{{}_{21}C_{3}} = \frac{6 \times 7}{1330} = \frac{42}{1330} \] Simplify the fraction by dividing numerator and denominator by 14: \[ \frac{42}{1330} = \frac{3}{95} \]

Final Answer