Explanation:
We have the following letters available:
A – 2, C – 1, I – 3, N – 1, O – 1, P – 2, R – 1, T – 2
Case 1: All 4 letters are different.
We have to select 4 letters out of 8 letters available. Number of ways = 8C2 = 28.
These 4 letters can be arranged in 4! = 24 ways.
∴ Total number of such words = 28 × 24 = 672.
Case 2: 2 letters are same and other 2 are different.
2 same letters can be chosen from either A, I, O or P i.e., in 4 ways.
2 different letters can be chosen from the remaining 7 letters in 7C2 = 21 ways.
∴ Total number of such selections = 4 × 21 = 84.
These 4 letters can be arranged in 4!/2! = 12 ways.
∴ Total number of such words = 84 × 12 = 1008.
Case 3: 2 same letters of 1 type and 2 same letters of other type
i.e., 2 pairs of letters can be chosen from either A, I, O or P in 4C2 = 6 ways.
These 4 letters can be arranged in 4!/(2!×2!) = 6 ways.
∴ Total number of such words = 6 × 6 = 36.
Case 4: 3 letters are same and 1 other letter is different.
3 same letters can only be I i.e., 1 way.
1 other letter can be chosen from remaining 7 letters i.e., 7C1 = 7 ways.
These 4 letters can be arranged in 4!/3! = 4 ways.
∴ Total number of such words = 4 × 7 = 28.
∴ Total number of ways 4 letter words can be formed = 672 + 1008 + 36 + 28 = 1744 ways.
Hence, 1744.