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Question: Prove that \[r!{}^n{C_r} = {}^n{P_r}\]...

Prove that r!nCr=nPrr!{}^n{C_r} = {}^n{P_r}

Explanation

Solution

Here in this question we must know the following permutation and combination formula. These are mentioned below: -
Permutation: - Number of permutation of ‘n’ things has taken ‘r’ at a time then permutation formula is given by: -
P(n,r)=nPrP(n,r) = {}^n{P_r}
P(n,r)=n!(n−r)!P(n,r) = \dfrac{{n!}}{{(n - r)!}}
Combination: -Number of combination of ‘n’ things has taken ‘r’ at a time then combination formula is given by: -
C(n,r)=nCrC(n,r) = {}^n{C_r}
C(n,r)=n!r!(n−r)!C(n,r) = \dfrac{{n!}}{{r!(n - r)!}}

Complete step-by-step answer:
We have to prove r!nCr=nPrr!{}^n{C_r} = {}^n{P_r}so we will simply apply the formula of permutation and combination.
L.H.S (left hand side) =r!nCrr!{}^n{C_r}
R.H.S (Right hand side) =nPr{}^n{P_r}
So we will try to convert the L.H.S side to the R.H.S side.
r!nCr=r!n!r!(n−r)!r!{}^n{C_r} = r!\dfrac{{n!}}{{r!(n - r)!}}
⇒n!(n−r)!\Rightarrow \dfrac{{n!}}{{(n - r)!}} (This is the formula of permutation)
Cancelling r!r! term from numerator and denominator.
∴n!(n−r)!=nPr\therefore \dfrac{{n!}}{{(n - r)!}} = {}^n{P_r}

Additional Information:
There are basically two types of permutation:
Repetition is Allowed: such as the possibilities of arrangement in passwords in lock.
No Repetition: for example the first three people in a running race. You can't be first and second. So here repetition cannot apply.
There are also two types of combinations (remember the order doesn't matter now):
Repetition is Allowed: such as coins in your pocket (5, 5, 5, 10, 10) so if this 5,5,5,10,10 is selected then it can be selected three times because repetition is allowed.
No Repetition: For example if you are having three balls 1, 2 and 3. They are chosen so possibility is only one 123 while in permutation 6 arrangements can be made For example 123, 321 ,132 ,213, 231 ,312

Note: Some students may find little confusion while memorising formula of permutation and combination so here is one tip in combination there isr!r! term extra in denominator while whole formula is same as permutation. Also permutation is the arrangement of items in which order matters while combination is the selection of items in which order does not matter.