Question

A={ 1,2,3,4,5,6} ; How many bijective functions \[f:A\to A\] have the property that \[f\left( 1 \right)\ne 2\] ?

Answer 1

Hint: To solve this particular problem and its related types, we need to have a basic as well as to some extent advanced knowledge of permutations and combinations as well as some basic multiplication techniques. When a set ‘A’ maps to itself then for the number of bijective functions there will be a permutation of the number of elements present in set ‘A’. However we are given a condition that \[f\left( 1 \right)\ne 2\] , so following this we can clearly explain that one of our mapping will get eliminated as it doesn’t satisfy the given condition.

Complete step by step solution:
Now we start off with the solution to the given problem by writing that,
The total number of elements present in set ‘A’ is equal to \[6\]. So the number of permutations for finding the number of bijective functions is equal to \[6!\] . Now according to the given condition we can clearly say that out of the \[6\] mappings one of them will not be satisfied. So the total number of functions will thus be equal to,
\[=6!\times \dfrac{5}{6}\] . This evaluated to \[=600\]

Note: For solving these and its related types of problems efficiently, we need to have a fair amount of ideas about permutations and combinations, an integral topic of algebra. We need to be careful while solving this problem about what all conditions that need to be met and how to implement the conditions in our problem. We also need to understand the terms bijection, mapping and functions which are very important to solve this problem quickly and efficiently.