The number of words that can be formed by using the letters of the word CALCULATE such that each word starts…
The number of words that can be formed by using the letters of the word CALCULATE such that each word starts and ends with a consonant, are
$5 \times 7 !$
$\frac{9 !}{8}$
$\frac{5 \times 7 !}{2}$
$20 \times 7 !$
Solution
Word CALCULATE has 9 letters.
Out of which ' $C$ ' repeats 2 times,
'A' repeats 2 times,
' $L$ ' repeats 2 times,
'E', 'U' and ' $T$ ' repeats once.
$\therefore \quad$ There are 5 consonants and 4 vowels.
Two consonants out of 5 can take start and end position of the word in ${ }^5 \mathrm{P}_2$ ways.
And remaining 7 letters can take remaining 7 positions in 7 ! ways.
Also, 'C', 'A' and ' $L$ ' repeats twice each.
$\therefore \quad$ The required number of words that can be formed $=\frac{5 p_2 \times 7 !}{2 ! \times 2 ! \times 2 !}=\frac{5 \times 4 \times 3 ! \times 7 !}{3 ! \times 2 \times 2 \times 2}=\frac{5 \times 7 !}{2}$