Finding n using Combination Formula

We are given the combination equation ${}^nC_4 = 70$. Our goal is to find the value of $n$.

Applying the Combination Formula

The formula for combinations is ${}^nC_r = \frac{n!}{r!(n-r)!}$.

Substituting $r=4$ into the formula, we get:

${}^nC_4 = \frac{n!}{4!(n-4)!}$

Solving the Equation

1. Set the formula equal to the given value:

   $\frac{n!}{4!(n-4)!} = 70$

2. Expand the factorial term $n!$ as $n \times (n-1) \times (n-2) \times (n-3) \times (n-4)!$:

   $\frac{n \times (n-1) \times (n-2) \times (n-3) \times (n-4)!}{4! \times (n-4)!} = 70$

3. Cancel out $(n-4)!$ from the numerator and denominator:

   $\frac{n(n-1)(n-2)(n-3)}{4!} = 70$

4. Calculate $4!$ which is $4 \times 3 \times 2 \times 1 = 24$:

   $\frac{n(n-1)(n-2)(n-3)}{24} = 70$

5. Multiply both sides by 24:

   $n(n-1)(n-2)(n-3) = 70 \times 24$

   $n(n-1)(n-2)(n-3) = 1680$

6. Find four consecutive integers whose product is 1680. We can test values starting from $n=5$ (since $r=4$, $n$ must be at least 4, and $n=4$ gives 0).
   • If $n=5$: $5 \times 4 \times 3 \times 2 = 120$ (Incorrect)
   • If $n=6$: $6 \times 5 \times 4 \times 3 = 360$ (Incorrect)
   • If $n=8$: $8 \times 7 \times 6 \times 5 = 1680$ (Correct)

Thus, the value of $n$ is 8.