Let $A = {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}}$ and $f : A \to A$ . The number of bijective functions such that $f (x_{i}) = x_{i}$ for exactly four of the $x_{i}$ 's is $(i = 1 to 6)$

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Solution

$4$ elements can be selected from $A$ in $^{6} C_{4}$ ways and they can be mapped such that $f (x_{i}) = x_{i}$ in $1$ way.

Since, we need exactly four to be mapped as $f (x_{i}) = x_{i}$ and the other $2$ should not be in the form of $f (x_{i}) = x_{i} .$
Hence, it has only $1$ possiblity.

$∴$ Required number of bijective functions $=^{6} C_{4} \times 1 = 15$

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