Question
The number of words that can be made by writing down the letters of the word CALCULATE such that each word starts and ends with a consonant, is
A.
$$\frac{{5\left( {7!} \right)}}{2}$$
B.
$$\frac{{3\left( {7!} \right)}}{2}$$
C.
$${2\left( {7!} \right)}$$
D.
None of these
Answer :
$$\frac{{5\left( {7!} \right)}}{2}$$
Solution :
The number of words like $$C - C = \frac{{7!}}{{2!\,2!}} = $$ the number of words like $$L - L.$$
The number of words beginning or ending with $$C,L = 2 \times \frac{{7!}}{{2!\,}}.$$
The number of words beginning or ending with $$\left( {C\,{\text{or }}L} \right),$$ $$T = 2\left( {2!} \right) \times \frac{{7!}}{{2!\,2!}}.$$
∴ the required number of words
$$\eqalign{
& = \frac{{7!}}{{2!\,2!}} + \frac{{7!}}{{2!\,2!}} + 2 \times \frac{{7!}}{{2!}} + 2\left( {2!} \right) \times \frac{{7!}}{{2!\,2!}} \cr
& = \left( {\frac{1}{4} + \frac{1}{4} + 1 + 1} \right)\left( {7!} \right). \cr} $$