Question

Number of all possible words (with or without meaning) that can be formed using all the letters of the word CABINET in which neither the word CAB nor the word NET appear is

• A. 5040
• B. 4806
• C. 4800
• D. 5034

Hint:

For problems involving restrictions like "these letters must be together" or "this pattern must not appear", the inclusion-exclusion principle is a powerful tool. Remember the formula for two sets: Total - (A or B) = Total - (A + B - A and B). When letters must be together, treat them as a single block for permutation purposes.

Answer 1

The Correct Option is B

Step 1: Total Permutations CABINET has 7 distinct letters. Total arrangements = \( 7! = 5040 \).
Step 2: Inclusion-Exclusion Principle Let \( A \) be the set of words containing "CAB" (treat as 1 unit). Remaining letters: I, N, E, T. Total entities: \{CAB\}, I, N, E, T (5 items). \( |A| = 5! = 120 \). Let \( B \) be the set of words containing "NET" (treat as 1 unit). Remaining letters: C, A, B, I. Total entities: \{NET\}, C, A, B, I (5 items). \( |B| = 5! = 120 \). Intersection \( A \cap B \): Contains both "CAB" and "NET". Entities: \{CAB\}, \{NET\}, I (3 items). \( |A \cap B| = 3! = 6 \).
Step 3: Calculation We need Total - \( |A \cup B| \). \( |A \cup B| = |A| + |B| - |A \cap B| = 120 + 120 - 6 = 234 \). Required Ways = \( 5040 - 234 = 4806 \).