Tag: exponent of a prime in n!

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Multiple choice exponent of a prime in n! factorial notation combinatorics and mathematical induction permutations and combinations maths

If $\displaystyle \frac{^{n}P _{r-1}}{a}=\frac{^{n}P _{r}}{b}=\frac{^{n}P _{r+1}}{c}$,then which of the following holds good 

  1. $c^{2}=a(b+c)$
  2. $a^{2}=c(a+b)$
  3. $b^{2}=a(b+c)$
  4. $\displaystyle \frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\displaystyle \frac{^{n}p _{r-1}}{a}=\frac{^{n}P _{r}}{b}$

$\displaystyle \Rightarrow n-r=\frac{b}{a}-1$

and $\displaystyle \frac{^{n}P _{r}}{b}=\frac{^{n}P _{r+1}}{c}$

$\displaystyle \Rightarrow n-r=\frac{c}{b}$
On dividing (i) and (ii) we get
$b^{2}=a(b+c)$