If the different permutations of all the letter of the word EXAMINATION are listed as in a dictionary, how many words are there in this list before the first word starting with E?
The word EXAMINATION contains 11 letters in which A appears 2 times, I appear 2 times and N appears 2 times. In dictionary, the letter that comes before E is A; thus, A is kept at the extreme left position by which the word begins. Then the arrangement will occur between the remaining 10 letters where there are 2 N and 2 I, which is the combination of 10 letters taken all at a time.
Therefore, the permutation is written as,
10 ! 2 ! 2 !
Cancel the common factors by factorizing the bigger term to the factorial.
The formula to calculate the factors of a factorial in terms of factorial itself is,
n ! = n ( n − 1 ) ! n ! = n ( n − 1 ) ( n − 2 ) ! [ n ≥ 2 ] ⋮
The term can be written as,
10 ! 2 ! 2 ! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 ! 2 ! 2 × 1 = 907200
Thus, the number of arrangement of the word is 907200.
The word EXAMINATION contains 11 letters in which A appears 2 times, I appear 2 times and N appears 2 times. In dictionary, the letter that comes before E is A; thus, A is kept at the extreme left position by which the word begins. Then the arrangement will occur between the remaining 10 letters where there are 2 N and 2 I, which is the combination of 10 letters taken all at a time.
Therefore, the permutation is written as,
Cancel the common factors by factorizing the bigger term to the factorial.
The formula to calculate the factors of a factorial in terms of factorial itself is,
The term can be written as,
Thus, the number of arrangement of the word is 907200.
