Find the number of words which can be formed by taking four letters at a time from the word “ COMBINATION”.

To determine the number of distinct words that can be formed by taking four letters at a time from the word "COMBINATION," we first need to analyze the letters in the word itself. The word "COMBINATION" consists of 11 letters, and some letters repeat. Specifically, the letters O and I each appear twice, while the others (C, M, B, N, T, A) appear just once.

Analyzing the letters

The unique letters in "COMBINATION" are: C, O, M, B, I, N, A, T. This gives us a total of 8 unique letters. However, since we have repetitions (O and I both appear twice), we need to consider different cases based on the number of repeated letters we include in our selections.

Cases to consider

• Case 1: Four different letters.
• Case 2: One letter appears twice, and two other different letters.
• Case 3: Two letters appear twice.

Calculating Case 1: Four different letters

We can select any 4 letters from the 8 unique letters. The number of ways to choose 4 from 8 can be calculated using the combination formula:

C(n, r) = n! / (r!(n - r)!)

Here, n = 8 (the number of unique letters) and r = 4 (the number of letters to choose).

Calculating this gives us:

C(8, 4) = 8! / (4! * (8 - 4)!) = 70

Each selection of 4 letters can be arranged in 4! (which is 24) ways, so the total for this case is:

70 * 24 = 1680

Calculating Case 2: One letter appears twice

In this scenario, we first select which letter will appear twice. We have two options: O or I. After selecting one of these letters, we then choose 2 more letters from the remaining 7 unique letters.

For each choice of the letter that appears twice, we can choose 2 from the remaining 7 letters:

C(7, 2) = 21

Since we can arrange the selected letters (including the repeated one) in a special way (4! / 2! = 12), the total number of arrangements for this case is:

2 * 21 * 12 = 504

Calculating Case 3: Two letters appear twice

In this case, we can only choose O and I, as they are the only letters that repeat. So we have one combination: O, O, I, I. The arrangement can be calculated as:

4! / (2! * 2!) = 6

This contributes 6 to our total count.

Summing it all up

Now, let's add up the totals from all three cases:

• Case 1: 1680
• Case 2: 504
• Case 3: 6

Total = 1680 + 504 + 6 = 2190

Therefore, the total number of distinct words that can be formed by taking four letters at a time from the word "COMBINATION" is 2190.