Question

The number of words that can be made by writing down the letters of the word $CALCULATE$ such that each word starts and end with a consonant, is

• A. $\frac{5}{2}(7!)$
• B. $\frac{3}{2}(7!)$
• C. $2(7!)$
• D. $3(7!)$

Answer 1

Answer: (A)

$\frac{5}{2}(7!)$

Given,
$CALCULATE$
there are $4$ vowels they are $A,A,U,E$ and $5$ consonants $C,C,L,L,T$
If there is no repeating consonant $(A,A),$ consonants should be placed at the starting and ending such that is can be arranged in ${}^5{\text{P}}_2$ ways then rest $3$ consonants and $4$ vowels in total $7$ can be arranged in $\frac{7!}{2!\times 2!}$ ways
but, there are $2A^{\prime }s,$
$\therefore$ Total number of ways are $\frac{{}^5{\text{P}}_2}{2!2!}\frac{7!}{2!}$
=$\frac{(7!)5}{2}$
Hence option A is correct