The permutations of n objects taken (i) atleast r at a time and (ii) atmost r at a time (if repetition of objects is allowed)is?

Dear Priyansh

when repetition is allowed then

 first object can be selected in = n ways

2nd object can be selected in = n ways

3rd  object can be selected in = n ways

.

.

rth  object can be selected in = n ways

so total ways  = n^r

for utmost r object number of ways = n⁰ + n¹ +n² ....n^r

for atleast r object is taken             = n^r + n^r+1 + .......+nⁿ

   find the sum of the series

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