The word TRIANGLE has 8 letters. We are required to find the number of ways in which the letters can be arranged so that the word angle is always present.


To solve this problem, we can use the technique of permutation and combination. We can first find the total number of ways in which the letters of the word TRIANGLE can be arranged, and then subtract the number of ways in which the word angle is not present.


The total number of ways in which the letters of the word TRIANGLE can be arranged is given by:
8! = 40320

The number of ways in which the word angle is not present can be calculated as follows:
- Remove the letters of the word angle from the word TRIANGLE. We are left with the letters T, R, I, N, and G.
- Arrange these letters in 5! ways.
- The letters of the word angle can be inserted into any of the 6 positions in the arrangement of the remaining letters (T, R, I, N, and G).
- Thus, the number of ways in which the word angle is not present is 5! x 6 = 720 x 6 = 4320.

Therefore, the number of ways in which the letters of the word TRIANGLE can be arranged so that the word angle is always present is given by:

40320 - 4320 = 36000


The number of ways in which the letters of the word TRIANGLE can be arranged so that the word angle is always present is 36000.