Question

How many ways are there to arrange the letters a, b, c, d, e, and f such that a is not directly followed by either b or c? For example, “abdef c” and “acdef b” are both invalid, but “adbcef” is valid.

Answer 1

Hi,

Please find the answer below:
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• Given: the letters a, b, c, d, e, and f
• Condition: a is not directly followed by either b or c

There are 6 letters.
The total number of ways letters can be arranged: 6! ways.
TOTAL = 720 ways.

Now we go opposite of what asked in the question condition.
Let's see how many ways a directly follows letter b. For that, we club
a and b -> 'ab' as one letter.

There are 5 letters. ab,c,d,e,f

These can be arranged in 5! ways.
Let's call it AB = 120 ways.

Let's see how many ways a directly follows letter c. For that, we club
a and c -> 'ac' as one letter.

There are 5 letters. b,ac,d,e,f

These can be arranged in 5! ways.
Let's call it AC = 120 ways.

The total number of ways letters can be arranged with the condition is:

TOTAL - { AB + AC }

= 720 - { 120 + 120 }

= 480 ways

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Hope this helps.