Question

The number of arrangement of 10 different things taken 4 at a time I which one particular thing always occur is

Answer 1

Since one particular thing will always occur , we have to arrange only 9 objects for 3 places i.e. P(9,3) = 60,480

Answer 2

Answer: 2016

Explanation:

The one particular thing that always occurs in the above case can be determined using permutation and combination.

The number of the arrangement of different things = 10

The number of things taken at a time = 4

The combination formula from a combination of r things in a set of n things:

$\mathrm{^nC_r}=\frac{\mathrm{n} !}{\mathrm{r} !(\mathrm{n}-\mathrm{r}) !}$

$^n C_r=\frac{^n P_r}{r !}$

Where,

$^nP_r$ is the permutation of n things taken at a time r

Applying the formula, we get,  

$^2 C_{3}=\frac{9 !}{3 !} \times 6=84$

With the fixed 4th thing, the number of ways for selecting that would be = 4! = 24

Required number of ways = 84 $\times$ 24 = 2016