Write the following products in factorial notation:

(i) Formula : $n !=n \times(n-1) \times(n-2) \ldots \ldots \ldots \ldots 3 \times 2 \times 1$

Let

$x=12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6$

Multiplying and dividing by $(5 \times 4 \times 3 \times 2 \times 1)$

$\therefore x=\frac{12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{5 \times 4 \times 3 \times 2 \times 1}$

From the above formula,

$x=\frac{12 !}{5 !}$

Conclusion :

$\therefore(12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6)=\frac{12 !}{5 !}$

(ii) Formula  $n !=n \times(n-1) \times(n-2) \ldots \ldots \ldots \ldots 3 \times 2 \times 1$

Let

$x=3 \times 6 \times 9 \times 12 \times 15$

Above equation can be written as

$x=3(1) \times 3(2) \times 3(3) \times 3(4) \times 3(5)$

$\therefore x=3^{5} \times(5 \times 4 \times 3 \times 2 \times 1)$

By using above formula,

$\therefore x=3^{5} \times(5 !)$

Conclusion :

$\therefore(3 \times 6 \times 9 \times 12 \times 15)=3^{5} \times(5 !)$