Question

How many words can be formed out of the letters of the word ARTICLE so that vowels occupy the even places?

Answer 1

Seven places : 3 even and 4 odd; 3 vow. and 4 cons.
${\therefore }^3{P}_3{\times }^4{P}_4=3!\times 4!=6\times 24=144$.
We have ten places out of which 5 places are old i.e. $1^{st},3^{rd},7^{th},9^{th}$ and five are even i.e. $2^{nd}$, fourth, sixth, eighth and tenth. In the five even places we have to fix up 4 vowels which can be done in ${}^5{P}_4$ ways. Having fixed up there vowels in even places, we will be left with six places namely 5 odd arid one even left after fixing the four vowels. In these six places we have to fix six consonants which can be done in ${}^6{P}_6$ i.e. $6!$ ways.
Thus the total number of ways is ${}^5{P}_4{\times }^6{P}_6$.
or $5!\times 6!=120\times 720=6(120{)}^2$.