Basic arithmetic

Question

I can't understand how: $$ \frac {2\times{^nC_2}}{5} $$

Equals:

$$ 2\times \frac {^nC_2}{5} $$

If we forget the combination and replace it with a $10$, the result is clearly different. $1$ in the first example and and $0.5$ in the second.

Answer 1

Remember some fact about fractions: $\dfrac{a}{b}\times \dfrac{c}{d} = \dfrac{ac}{bd} \Rightarrow \dfrac{2\times \binom{n}{2}}{5} = \dfrac{2\times \binom{n}{2}}{1\times 5} = \dfrac{2}{1}\times \dfrac{\binom{n}{2}}{5} = 2\times \dfrac{\binom{n}{2}}{5}$

Answer 2

$$\frac{a}{b}\times\frac{c}{d}=\frac{a\times c}{b \times d}$$ So, taking your example $$\frac{2\times10}{5\times 1}=\dfrac{2}{5}\times\dfrac{10}{1}=\dfrac{2}{1}\times\frac{10}{5}=\dfrac{10}{5}\times 2=\dfrac{2}{5}\times 10$$ As multiplication is commutative.

Answer 3

Using your example, $$ \frac{2\times 10}{5}=\frac{20}{5}=4$$ And $$ 2\times\frac{10}{5}=2\times 2=4$$ In general for any $c\not =0$, we have $$ \frac{a\times b}{c}=a\times\frac{b}{c}= b\times\frac{a}{c} $$