If all the permutations of the letters of the word TACKLE are written in order as in a dictionary, then digit in unit place of the rank of the word TACKLE is

We are asked to find the digit in the unit place of the rank of the word "TACKLE" when all its permutations are arranged in dictionary order.

Step 1: Identify the Letters

The word "TACKLE" consists of the letters: T, A, C, K, L, E. These letters are all distinct.

Step 2: Arranging Letters in Lexicographical Order

The lexicographical order of the letters is:

A, C, E, K, L, T

Step 3: Finding the Position of "TACKLE"

We now calculate the number of words that come before "TACKLE" in dictionary order. We examine each letter in the word "TACKLE" one by one and count the permutations that come before it.

First Letter: T

The first letter in "TACKLE" is T. We count how many words begin with a letter smaller than T.

The letters smaller than T are A, C, E, K, and L, making 5 options for the first letter. Each of these 5 letters can be followed by any permutation of the remaining 5 letters, giving 5 × 120 = 600 words.

Second Letter: A

The second letter in "TACKLE" is A. Since A comes before T, we don’t need to count any words here.

Third Letter: C

The third letter in "TACKLE" is C. We count how many words begin with "TA" and then have a letter smaller than C in the third place.

The letters smaller than C (after T) are A, which gives 1 option. This 1 option can be followed by any permutation of the remaining 4 letters, giving 1 × 24 = 24 words.

Fourth Letter: K

The fourth letter in "TACKLE" is K. We count how many words begin with "TAC" and have a letter smaller than K in the fourth place.

The letters smaller than K (after TAC) are E and K, giving 2 options. Each of these can be followed by any permutation of the remaining 3 letters, giving 2 × 6 = 12 words.

Fifth Letter: L

The fifth letter in "TACKLE" is L. We count how many words begin with "TACK" and have a letter smaller than L in the fifth place.

The letter smaller than L (after TACK) is E, giving 1 option. This option can be followed by any permutation of the remaining 2 letters, giving 1 × 2 = 2 words.

Sixth Letter: E

The sixth letter in "TACKLE" is E. As we are already at the last letter of the word "TACKLE", no words are counted here.

Step 4: Calculating the Rank

To find the total number of words that come before "TACKLE", we sum up the number of words calculated in each step:

600 + 24 + 12 + 2 = 638

So, the rank of the word "TACKLE" is 639 (since we include "TACKLE" itself).

Step 5: Final Answer

The digit in the unit place of the rank 639 is 9.