Find the number of words formed by permuting all the letters
of the word INDEPENDENCE such that the E's do not come together.
Use the "backdoor" approach,
that is
find the number of ways when they are together.
then we have EEEE, I, N,N,N,D,D,P,C
arrange 9 things, with 3N's and 2 D's
= 9!/(3!2!)
= 30240
number of arrangements without restriction
= 12!/(4!3!2!) = 1663200
so number of ways when they are not together
= 1663200 - 30240 = 1632960
great answers
To find the number of words formed by permuting all the letters of the word "INDEPENDENCE" such that the E's do not come together, we can use the concept of permutations with restrictions.
Step 1: Find the total number of permutations.
The word "INDEPENDENCE" has a total of 13 letters. To find the total number of permutations, we can use the formula for permutations of distinct objects, which is n!.
13! = 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 6,227,020,800
Therefore, there are 6,227,020,800 possible permutations of the word "INDEPENDENCE" without any restrictions.
Step 2: Find the number of permutations with the E's together.
To find the number of permutations where the E's come together, we can treat the two E's as a single unit. So, we can consider "EEP" as one unit. With this, the total number of units we have is now 11 (10 original units + 1 unit for the E's).
Now, we find the total number of permutations of these 11 units. Using the formula for permutations of n objects with m indistinguishable objects, the number of permutations can be calculated as:
11! / (2! x 1!)
= (11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) / (2 x 1 x 1)
= 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3
= 95,040
Therefore, there are 95,040 permutations where the E's come together.
Step 3: Find the number of permutations where the E's do not come together.
To find the number of permutations where the E's do not come together, we subtract the number of permutations where the E's come together from the total number of permutations.
Total number of permutations - Number of permutations with E's together
= 6,227,020,800 - 95,040
= 6,226,925,760
Therefore, there are 6,226,925,760 words formed by permuting all the letters of the word "INDEPENDENCE" such that the E's do not come together.