Find the number of selections of six letters from the letters of the word 'K A R N A T A K A'

Answer

To find the number of selections of six letters from the letters of the word "KARNATAKA", we can follow these steps:

### Step 1: Identify the letters and their frequencies
The letters in the word "KARNATAKA" are:
- K: 3 times
- A: 4 times
- R: 1 time
- N: 1 time
- T: 1 time

### Step 2: Determine the total number of letters
The total number of letters in "KARNATAKA" is 10.

### Step 3: Consider different cases for selecting 6 letters
Since we have repeated letters, we need to consider different cases based on how many times we select each letter. We will break it down by the number of A's selected.

#### Case 1: 4 A's
- **Sub-case 1.1**: 4 A's and 2 K's
- Selection: 4 A's, 2 K's
- This is one valid selection.

- **Sub-case 1.2**: 4 A's and 1 letter from {R, N, T}
- We can choose 1 letter from the remaining 3 letters (R, N, T).
- Number of ways = 3C1 = 3

#### Case 2: 3 A's
- **Sub-case 2.1**: 3 A's and 3 letters from {K, R, N, T}
- We can choose 3 letters from the remaining 4 letters (K, R, N, T).
- Number of ways = 4C3 = 4

- **Sub-case 2.2**: 3 A's and 2 K's and 1 letter from {R, N, T}
- We can choose 1 letter from the remaining 3 letters (R, N, T).
- Number of ways = 3C1 = 3

#### Case 3: 2 A's
- **Sub-case 3.1**: 2 A's and 4 letters from {K, R, N, T}
- We can choose 4 letters from the remaining 4 letters (K, R, N, T).
- Number of ways = 4C4 = 1

- **Sub-case 3.2**: 2 A's and 2 K's and 2 letters from {R, N, T}
- We can choose 2 letters from the remaining 3 letters (R, N, T).
- Number of ways = 3C2 = 3

#### Case 4: 1 A
- **Sub-case 4.1**: 1 A and 5 letters from {K, R, N, T}
- This case is not possible since we only have 4 different letters left.

### Step 4: Summarize the counts from each case
Now we can summarize the counts from the valid cases:
- From Case 1: 1 (4 A's, 2 K's) + 3 (4 A's, 1 from {R, N, T}) = 4
- From Case 2: 4 (3 A's, 3 from {K, R, N, T}) + 3 (3 A's, 2 K's, 1 from {R, N, T}) = 7
- From Case 3: 1 (2 A's, 4 from {K, R, N, T}) + 3 (2 A's, 2 K's, 2 from {R, N, T}) = 4

### Step 5: Calculate the total number of selections
Total selections = 4 (from Case 1) + 7 (from Case 2) + 4 (from Case 3) = 15

### Final Answer
The total number of selections of six letters from the letters of the word "KARNATAKA" is **15**.