A secure access code is formed by arranging all 6 of the distinct characters A, E, B, C, D… — GMAT

A secure access code is formed by arranging all $6$ of the distinct characters $A, E, B, C, D,$ and $F$ in a single line. If the two vowels ( $A$ and $E$ ) cannot be adjacent to each other, and the code cannot begin with the consonant $B$ , how many different valid access codes can be formed?

A
240
B
252
C
360
408Answer
E
480

Answer

408 valid access codes can be formed.

To find the number of valid arrangements satisfying both conditions, we use complementary counting. First, the total number of arrangements of 6 distinct characters is 6! = 720. Next, we determine the number of invalid arrangements. Let V be the set of arrangements where A and E are adjacent: treating (AE) as a single unit gives 5! × 2! = 240 ways. Let B_first be the set of arrangements starting with B: fixing B in the first spot leaves 5! = 120 ways. The intersection of these two invalid sets (where A and E are adjacent AND the code starts with B) places B in position 1 and arranges the remaining 4 items including the (AE) block in 4! × 2! = 48 ways. By inclusion-exclusion, the total invalid arrangements equal 240 + 120 - 48 = 312. Subtracting invalid arrangements from the total gives 720 - 312 = 408 valid access codes.

Step-by-Step Solution

Calculate the total total number of unrestricted linear arrangements of the 6 distinct characters.

6! = 720 total arrangements.

6 distinct characters can be ordered in 6! different ways.

Calculate the number of arrangements where the two vowels (A and E) ARE adjacent (Set V).

|V| = 240 arrangements.

Treating (AE) as 1 block leaves 5 items to arrange in 5! = 120 ways. Since (AE) has 2! = 2 internal arrangements, |V| = 2 * 120 = 240.

Calculate the number of arrangements that BEGIN with the consonant B (Set B).

|B| = 120 arrangements.

Fixing B in the first position leaves 5 remaining positions to be filled by the remaining 5 characters in 5! = 120 ways.

Calculate the overlap: arrangements where vowels ARE adjacent AND the code BEGINS with B (Set V ∩ B).

|V ∩ B| = 48 arrangements.

With B fixed in position 1, the remaining 5 positions are filled by 4 blocks: (AE), C, D, F. These 4 blocks can be arranged in 4! = 24 ways, and (AE) has 2! = 2 internal orderings, giving 24 * 2 = 48.

Apply the Principle of Inclusion-Exclusion to find the total invalid arrangements, and subtract from the total.

720 - (240 + 120 - 48) = 720 - 312 = 408 valid arrangements.

The number of invalid arrangements is |V ∪ B| = |V| + |B| - |V ∩ B| = 312. Subtracting invalid codes from total codes yields 408.

Key Concept

Linear permutations with multiple restrictions using complementary counting and inclusion-exclusion.

Estimated Time:2m 0s

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