Question

Number of words that can be formed using all the letters of the word GARGEE if no two alike letters are together, is

• A. 84
• B. 62
• C. 240
• D. 280

Answer 1

Answer: (A)

Total $-n(A\cup B)=\frac{6!}{2!2!}-(n(A)+n(B)-n(A\cap B))$
$=\frac{6!}{2!2!}-\left(\frac{5!}{2!}+\frac{5!}{2!}-4!\right)=180-96=84$

Set $A$ represents number of ways when $G^{\prime }s$ are together
Set B represents number of ways then E's are together
Aliter: GGEEAR
Number of words when G's are separated
$=\frac{4!}{2!}\cdot {}^5{C}_2=120$
Number of words when G's are separated but E's are together $=3!\times {}^4{C}_2=36$
$\therefore$ Number of ways when no two alike letters are together
$=120-36=84$