Question

The number of permutations of the letters of the word AGAlN taken three at a time is

• A. 48
• B. 24
• C. 36
• D. 33

Answer 1

The correct option is D 33

Number of letters in the word A G A l N are $=$ 5 out of which 2A, 1G, 1N, 1I $\therefore$Number of permutations taken three at a time $=$ Coefficient of $x^3$ in $3!(1+\frac{x}{1!}+\frac{x^2}{2!})(1+x{)}^3$

$=$ Coefficient of $x^3$ in $\left(\frac{6(2+2x+x^2}{2}\right)(1+x{)}^3$ $=$ Coefficient of $x^3$ in $3\{1+(1+x{)}^2\}(1+x{)}^3$ $=$ Coefficient of $x^3$ in $3\left\{\right(1+x{)}^3+(1+x{)}^5\}$ $=3\left\{{}^3{C}_3{+}^5{C}_3\right\}=3(1+10)=33$

Hence, (d) is correct answer.