JEE Main 2022 · Mathematics · Sets, Relations and Functions · hard

The number of one-one functions f:\1,2,3,4\→\1,2,…,7\ such that f(i)≠ i for every i∈\1,2,3,4\ is

Single correct option

The number of one-one functions

f:\{1,2,3,4\}\to\{1,2,\dots,7\}

such that

f(i)\ne i

for every

i\in\{1,2,3,4\}

Show answer and solution

Answer: (C) $465$

Use inclusion-exclusion on the bad events

E_i:\ f(i)=i

. The total number of one-one maps is

^7P_4=840

. If

k

chosen elements are fixed, the remaining

4-k

elements map injectively into the other

7-k

values, giving

^{7-k}P_{4-k}

maps. So the count is

840-4\cdot{}^6P_3+6\cdot{}^5P_2-4\cdot{}^4P_1+1=840-480+120-16+1=465

. Stopping before the last alternating terms gives the nearby wrong values

480

464

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