Question

Find WITHOUT assuming any formula, the permutations of $35$ different items taken $13$ at a time. (answer is factorial notation ).

Answer 1

1st item can be taken in 35 ways....

2nd item can be taken in 34 ways...

3rd item can be taken in 33 ways...

similarly.....

.

.

13th item can be taken in 23 ways...

so permutation of 35 items taken 13 at a time=$35\ast 34\ast 33\ast \cdot \cdot \cdot \cdot \cdot \cdot \cdot \ast 23$

multiplying 22! both in numerator and denominator we get

=$\frac{35\ast 34\ast 33\ast \cdot \cdot \cdot \cdot \cdot \cdot \cdot \ast 23\ast 22!}{22!}$

=$\frac{35!}{22!}$ (ans)