Question

In the word "ENGINEERING" if all E's are not together and N's come together then number of permutations is

Answer 1

Solution:

Number of Alphabets in the word ENGINEERING= 11

E→3

N→3

G→2

I→2

R→1

Permutation of the word ENGINEERING= $\frac{11!}{3! \times 3!  \times 2! \times 2! \times 1!}=\frac{11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3!}{3! \times 6  \times 4}=11 \times 10 \times 9 \times 8 \times 7 \times 5=990 \times 280=277200$

Now considering 3 Alphabet E and three Alphabet N together they can be arranged in $\frac{6!}{3! \times 3!}= 20$ ways

Number of permutation of word "ENGINEERING" if all E's are not together and N's come together = Permutation of the word ENGINEERING -Permutation of the word ENGINEERING when all E's and N's come together

= 277200 - [Considering alphabet N and E as single alphabet the remaining alphabets are GGIINNR, so if you will see 8 places are empty between each alphabet, so you can place three N and three E at any of 8 places ]

= 277200 - $8 \times\frac{6!}{3!\times 3!} \times \frac{7!}{2!\times 2!\times2!}$

= 277200 - 8 × 20× 630

= 277200 - 100800

= 176400