Permutations and Combinations - Factorial notation

Evaluate the expression: $\frac{8!}{6! \times 2!}$

We can write $8!$ as $8 \times 7 \times 6!$ to simplify the expression: $\frac{8 \times 7 \times 6!}{6! \times 2 \times 1}$ Cancelling $6!$ from the numerator and denominator: $\frac{8 \times 7}{2 \times 1} = \frac{56}{2} = 28$

To evaluate fractions involving factorials, expand the larger factorial until it matches the largest factorial in the denominator to simplify calculations.

Find $x$ if $\frac{1}{8!} + \frac{1}{9!} = \frac{x}{10!}$

Write all terms with $8!$ in the denominator: $\frac{1}{8!} + \frac{1}{9 \times 8!} = \frac{x}{10 \times 9 \times 8!}$ Multiply the entire equation by $8!$: $1 + \frac{1}{9} = \frac{x}{90}$ $\frac{10}{9} = \frac{x}{90}$ $x = \frac{10 \times 90}{9} = 100$

In equations involving factorials, it is efficient to express all factorials in terms of the smallest factorial appearing in the equation.

Compute $\frac{n!}{(n-r)!}$ when $n = 6$ and $r = 2$.

Substituting the values of $n$ and $r$: $\frac{6!}{(6-2)!} = \frac{6!}{4!}$ Expanding $6!$: $\frac{6 \times 5 \times 4!}{4!} = 6 \times 5 = 30$

Subtract the values in the denominator first, then expand the numerator's factorial to cancel out the denominator.