Q41 / 160 · AP EAMCET 2023 (17 May Shift 1) · Mathematics

If a set $\mathrm{A}$ has $\mathrm{n}$ elements, then the number of functions defined from A to A that are not one-one is

  • A
    $(n)^{n^2}$
  • B
    $\mathrm{n} !-\left({ }^{\mathrm{n}} \mathrm{C}_0+{ }^{\mathrm{n}} \mathrm{C}_1+{ }^{\mathrm{n}} \mathrm{C}_2+\ldots+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}}\right)$
  • Correct answer
    C
    $\mathrm{n}^{\mathrm{n}}-\mathrm{n} !$
  • D
    $\mathrm{n}^{\mathrm{n}}$

Solution

No. of functions from $A$ to $A=n^n$ No. of one-one functions $=\frac{n !}{(n-n) !}=n !$ $\therefore \quad$ No. of functions defined from $A$ to $A$ that are not oneone $=n^n-n !$

Hence, the correct answer is option: 'C'.

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