Permutations And Combinations

The word ARRANGEMENT} consists of 11 letters:
A, R, R, A, N, G, E, M, E, N, T
Frequencies of repeating letters:
A – 2, R – 2, N – 2, E – 2, G – 1, M – 1, T – 1
We are to form 9-letter words using these letters, taking into account the repeated letters.
To count the number of distinct 9-letter words, we need to consider all valid combinations of 9 letters and for each, count permutations accounting for repetition.
One of the valid combinations (which contributes the maximum) is:
Selecting A, R, R, N, N, E, E, G, M
Here, R, N, and E each appear twice. Total letters = 9
So, the number of such arrangements is:
There are multiple such valid selections. When all such cases are computed and their results summed, the total count is:
Thus, the total number of distinct 9-letter words that can be formed is 302400.

To form a three-digit number, we have three places to fill: hundreds, tens, and ones. The possible digits are 0, 1, 2, 3, and 5, but the first digit (hundreds place) cannot be 0 (since it is a three-digit number). So, we analyze the choices for each place:

Step 1: Hundreds Place.
The hundreds place can be filled by any digit except 0, so we have the choices 1, 2, 3, or 5. Hence, there are 4 choices for the hundreds place.

Step 2: Tens Place.
The tens place can be filled by any of the 5 digits (0, 1, 2, 3, 5), so there are 5 choices for the tens place.

Step 3: Ones Place.
The ones place can also be filled by any of the 5 digits (0, 1, 2, 3, 5), so there are 5 choices for the ones place.

Step 4: Total Combinations.
The total number of three-digit numbers that can be formed is the product of the number of choices for each place: Thus, the number of such three-digit numbers is 100.