Permutations and Combinations

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 $1(\mathrm{Cl})$ : 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:CRICEKCRICKET (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.