Question

Find the number of arranging 11 distinct objects taken 4 at a time so that a specified object always occurs.

Answer 1

There are 11 distinct objects and 4 objects are arranged at a time.
The number of permutations of n distinct objects, taken r at a time, when one specified object will always occur is `"r"xx^(("n"-1))"P"_(("r"-1))"`
Here, r = 4, n = 11
∴ The number of permutations of 4 out of 11 objects when a specified object occurs.
= `4xx^((11-1))"P"_((4-1))=4xx^10"P"_3`

 = `4xx(10!)/((10-3)!)`

 = `4xx(10!)/(7!)`

= `4xx(10xx9xx8xx7!)/(7!)`

 = 2880
∴ There are 2880 permutations of 11 distinct objects, taken 4 at a time, in which one specified object always occurs.