Question

The number of different four-digit numbers which can be made out using one $1$, two $2^{\prime }s$, three $3^{\prime }s$ and four $4^{\prime }s$, is

• A. $192$
• B. $126$
• C. $164$
• D. $175$

Answer 1

The correct option is D $175$

Case $\left(1\right):$ All $4$ alike $=1$

Case $\left(2\right):3$ alike $+1$ different
$({}^2{C}_1\times {}^3{C}_1)\times \frac{4!}{3!}=24$

Case $\left(3\right):2$ alike $+2$ different $({}^3{C}_1\times {}^3{C}_2)\times \frac{4!}{2!}=108$

Case $\left(4\right):2$ alike $+2$ other alike${}^3{C}_2\times \frac{4!}{2!2!}=18$

Case $\left(5\right):$ All $4$ different
$1\times 4!=24$

Total $=1+24+108+18+24=175$