Find the number of arrangements of the letter | vedclass

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Question

(A) The word $INDEPENDENCE$ has $12$ letters: $I(1), N(3), D(2), E(4), P(1), C(1)$.There are $5$ vowels: $E, E, E, E, I$.Treating the $5$ vowels as a

Vedclass · Accepted Answer

$16800$

Vedclass · Answer

(A) The word $INDEPENDENCE$ has $12$ letters: $I(1), N(3), D(2), E(4), P(1), C(1)$.There are $5$ vowels: $E, E, E, E, I$.Treating the $5$ vowels as a single object,we have $7$ consonants $(N, N, N, D, D, P, C)$ plus $1$ vowel-block,totaling $8$ objects.The number of ways to arrange these $8$ objects,where $N$ repeats $3$ times and $D$ repeats $2$ times,is $\frac{8!}{3!2!}$.The $5$ vowels $(E, E, E, E, I)$ can be arranged among themselves in $\frac{5!}{4!}$ ways.By the multiplication principle,the total number of arrangements is $\frac{8!}{3!2!} 	imes \frac{5!}{4!} = \frac{40320}{6 	imes 2} 	imes 5 = 3360 	imes 5 = 16800$.

Find the number of arrangements of the letters of the word $INDEPENDENCE$. In how many of these arrangements do all the vowels always occur together?

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