Question

Let A = {1, 2, 3, ...n} and B = {a, b}. Then the number of onto functions from A into B is

• A. $nP2$
• B. $2^n-1$
• C. $2^n-2$
• D. $\left(n-1\right)!$

Answer 1

The correct option is C $2^n-2$

$\mathrm{Given}\left(A\right|=n\mathrm{and}\left(B\right|=2\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{functions}\mathrm{from}A\mathrm{to}B=2^n\mathrm{Consider}\mathrm{the}\mathrm{constant}\mathrm{functions}f\left(x\right)=a\mathrm{for}\mathrm{all}x\in A\mathrm{and}g\left(x\right)=b\mathrm{for}\mathrm{all}x\in A\mathrm{These}\mathrm{two}\mathrm{functions}\mathrm{are}\mathrm{into},i.e.,\mathrm{they}\mathrm{are}\mathrm{not}\mathrm{onto}.\therefore \mathrm{Number}\mathrm{of}\mathrm{onto}\mathrm{functions}\mathrm{from}A\mathrm{to}b\mathrm{is}{2}^n-2.$