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  • State whether the statements in True or False. In the permutations of n things, r taken together, the number of permutations in which m particular things occur together is n–mPr–m × rPm.

State whether the statements in True or False.

In the permutations of n things, r taken together, the number of permutations in which m particular things occur together is ^{n-m}P_{r-m} \times ^rP_m.
 

Answers (1)

False. In  arrangement of n things taken r at a time in which m occur together, first we

 select \left ( r-m \right ) objects from \left (n-m \right ) objects in  ^{n-m}C_{r-m} ways.

 If we consider these m things as 1 group number of objects excluding these m objects=\left ( r-m \right )

  Now, we have to arrange\left ( r-m+1 \right ) objects 

 Number of arrangements=\left ( r-m+1 \right )!

Also, m objects which are considered as 1 group can be arranged in m!ways  

 Required number of arrangements=  ^{n-m}C_{r-m}*\left (r-m+1! \right )*m!  

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