Question

Total number of distinct 4 letters words that can be formed from the letters of the word 'NANDINI', so that at most 2 alike letters are together, is

• A. 104
• B. 106
• C. 108
• D. 114

Answer 1

Answer: (C)

108

Answer 2

The word 'NANDINI' has the following letters: N (3 times), A (1 time), D (1 time), I (2 times). We need to form distinct 4-letter words such that at most 2 alike letters are together.

Case 1: All 4 letters are distinct. The distinct letters are N, A, D, I. The number of permutations is $4! = 24$. These words are valid.

Case 2: Exactly 2 letters are alike, and the other 2 are distinct (XXYZ form).

• Subcase 2.1: Pair is NN. We choose 2 distinct letters from {A, D, I} in $\binom{3}{2} = 3$ ways. The sets are {N, N, A, D}, {N, N, A, I}, {N, N, D, I}. For each set, the number of arrangements is $\frac{4!}{2!} = 12$. Total arrangements = $3 \times 12 = 36$. These words are valid.
• Subcase 2.2: Pair is II. We choose 2 distinct letters from {N, A, D} in $\binom{3}{2} = 3$ ways. The sets are {I, I, N, A}, {I, I, N, D}, {I, I, A, D}. For each set, the number of arrangements is $\frac{4!}{2!} = 12$. Total arrangements = $3 \times 12 = 36$. These words are valid. Total for Case 2 = $36 + 36 = 72$.

Case 3: Exactly 2 letters are alike, and another 2 letters are alike (XXYY form). The only possible set is {N, N, I, I}. The number of distinct arrangements is $\frac{4!}{2!2!} = \frac{24}{4} = 6$. These words are valid.

Case 4: Exactly 3 letters are alike, and the other 1 is distinct (XXXY form). The only possibility is NNNX, where X is chosen from {A, D, I} in $\binom{3}{1} = 3$ ways. The sets are {N, N, N, A}, {N, N, N, D}, {N, N, N, I}. For each set, total arrangements are $\frac{4!}{3!} = 4$. We must subtract arrangements where NNN are together. For each set, there are 2 such arrangements (e.g., NNNA, ANNN). So, valid arrangements per set = $4 - 2 = 2$. Total for Case 4 = $3 \times 2 = 6$.

Case 5: All 4 letters are alike (XXXX form). This is not possible as no letter appears 4 times.

Total Number of Distinct Words: Total = Case 1 + Case 2 + Case 3 + Case 4 Total = $24 + 72 + 6 + 6 = 108$.