The word EVERGREEN can be arranged in the following manner.
What is Arrangement?
The number of possible arrangements or orders for a set of items is known as an arrangement number, also known as a permutation number or just a permutation.
- Step 1: When there are no restrictions
The letter EVERGREEN has 9 letter. So, the total arrangements will be 9!.
But letter 'E', 'R' are repeating 4 times and 2 times respectively.
Thus, total arrangements becomes 9!/(4!×2!) = 7560.
- Step 2: When the first letter is R and the last letter is G
As, first and last letter are fixed for R and g respectively.
Thus the total positions left are 7! with 'E', 'R' are repeating 4 times and 2 times respectively.
Thus, total arrangements becomes 7!/(4!×2!) = 105.
- Step 3: When the Es are all together. Three letters from the 9 letters of the word EVERGREEN are selected.
The total combination for arrangement is 7560.
But when all 4 E's are kept together its gives 4! = 24.
Therefore, the total combination becomes 7560×120 = 181440.
- Step 4: When the number of selections which contain no Es and exactly 1 R.
As, no E and 1 R is not to be taken which left 5 positions with no repetitions of letter
Thus, the total combination becomes 4! = 24.
- Step 5: The number of selections which contain no Es.
The total letter left after the removal of all 4 E's is 5 with times repetition of R.
Therefore, the total combination will be 5!/2! = 60.
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