Question

What does $nP_r $ and $nC_r$ mean?

Answer 1

Hint: Here, we are going to explain two concepts ($n\Pr $and $nCr$) which are straightforward and should not take too long to explain. I am going to explain your differences; when to use them, whenever you are trying to find the total number of outcomes in a given situation. For example, if you have 3 pants and 4 shirts. Then by using a simple calculation, you would have $3\times 4=12$ total outfits.

Complete step by step solution:
 In mathematics, permutation refers to the arrangement of all the members of a set in some order or sequence, while combination does not regard the order as a parameter. It is just a way of selecting items from a set or collection.
 A permutation is the arrangements of $r$ things from a set of $n$ things without replacement. Order matters in the permutation.
${{\Rightarrow }_{n}}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}$
A combination is the choice of $r$things from a set of $n$things without replacement. The order does not matter in combination.
${{\Rightarrow }_{n}}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$
In both above formulas the factorial (!) means it is a function that multiplies a number by every number below it. For example, $5!=5\times 4\times 3\times 2\times 1$.
Example: Find the number of permutations of the letters of the word ‘REMAINS’ such that the vowels always occur in odd places.