Question

How many words can be formed by taking 4 letters at a time from the letters of the word 'MORADABAD'?

Answer 1

$$\begin{gathered} \mathrm{In}\mathrm{MORADABAD}\mathrm{total}\mathrm{letters}=9.\mathrm{In}\mathrm{which}\mathrm{there}\mathrm{are}3\mathrm{A}\text{'}\mathrm{s},2\mathrm{D}\text{'}\mathrm{sand}4 \\ \mathrm{different}\mathrm{letters}. \\ \mathrm{Case}-1: \\ \mathrm{No}.\mathrm{of}\mathrm{words}\mathrm{formed}\mathrm{from}4\mathrm{different}\mathrm{letters}, \\ {}^6\mathrm{C}_4\times 4! \\ =15\times 24=360 \\ \mathrm{Case}-2: \\ \mathrm{No}.\mathrm{of}\mathrm{words}\mathrm{formed}\mathrm{from}2\mathrm{alike}\mathrm{and}2\mathrm{different}\mathrm{letters}, \\ ={}^2\mathrm{C}_1\times {}^5\mathrm{C}_2\times \frac{4!}{2!} \\ =2\times 10\times 12=240 \\ \mathrm{Case}-3: \\ \mathrm{No}.\mathrm{of}\mathrm{words}\mathrm{formed}\mathrm{from}3\mathrm{alike}\mathrm{and}1\mathrm{different}\mathrm{letters}, \\ =1\times {}^5\mathrm{C}_1\times \frac{4!}{3!}=5\times 4=20 \\ \mathrm{Case}-4: \\ \mathrm{No}.\mathrm{of}\mathrm{words}\mathrm{formed}\mathrm{from}2\mathrm{alike}\mathrm{of}\mathrm{one}\mathrm{kind}\mathrm{and}2\mathrm{alike}\mathrm{of}\mathrm{other}\mathrm{kind}, \\ ={}^2\mathrm{C}_2\times \frac{4!}{2!\times 2!}=1\times 6=6 \\ \mathrm{So}\mathrm{number}\mathrm{of}\mathrm{required}\mathrm{words}=360+240+20+6=626 \end{gathered}$$