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