Solution:
To find the rank of the word "CRICKET" when all its permutations are arranged in dictionary order, follow these steps:
List the Letters in Alphabetical Order: Arrange the letters of "CRICKET" alphabetically: C, C, E, I, K, R, T.
Compute Permutations for Each Case:
Starting with C followed by C (CC): Calculate permutations of the remaining letters E, I, K, R, T. This would be 5 ! = 120 ways.
Starting with C followed by E(CE): Calculate permutations of the remaining letters C, I, K, R, T. This would also be 5!=120 ways.
Starting with C followed by 1(Cl) : Calculate permutations of the remaining letters C,E,K,R,T. Again, 5!=120 ways.
Starting with C followed by K (CK): Calculate permutations of the remaining letters C, E, I, R, T. Again, 5 ! = 120 ways.
Starting with C followed by R and then C (CRC): Calculate permutations of the remaining letters E,I,K,T. This is 4!=24 ways.
Starting with C followed by R and then E (CRE): Calculate permutations of the remaining letters C,I,K,T. This also results in 4!=24 ways.
Starting with CRIC: You need to arrange the remaining letters K,E,T. "CRICE" results in 2 distinct permutations:
CRICEK
CRICKET (the word we are interested in)
Therefore, the number of arrangements before CRICKET is 2 (starting at CRICE_).
Determine the Rank:
The rank of "CRICKET" is therefore 120+120+120+120+24+24+2+1=531.
Thus, the rank of the word "CRICKET" is 531 in the list of all possible permutations arranged in dictionary order.
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