JEE Main 2019 (Online) 11th January Evening ShiftQuestion 1
The number of functions $f$ from $\{1,2,3,\ldots,20\}$ onto $\{1,2,3,\ldots,20\}$ such that $f(k)$ is a multiple of $3$, whenever $k$ is a multiple of $4$ is :
Permutation and Combination
Solution
Concept. An onto map from a finite set to itself is a bijection. Fill the restricted elements first. Formulas used. (1) ${}^nP_r=\frac{n!}{(n-r)!}$Step 1 - Restricted inputs. Multiples of $4$: $4,8,12,16,20$ ($5$ numbers). Step 2 - Allowed images. Multiples of $3$: $3,6,9,12,15,18$ ($6$ numbers). Step 3 - Map the 5 inputs. Distinct images from $6$: ${}^6P_5=\frac{6!}{1!}=6!$. Step 4 - Remaining inputs. $15$ inputs onto the $15$ unused values: $15!$. Step 5 - Total. $15!\times6!$. Answer: C