Consider counting the number of orderings of the 26 letter E

Consider counting the number of orderings of the 26 letter English alphabet which do not include the words AMBER, BERTH, or THINK.

Let U be the universe of all orderings of the alphabet, so |U| = 26!

Define c₁ to be the condition that AMBER appears, c₂ the condition that BERTH appears, and c₃ the condition that THINK appears. Complete each of the following. (Use factorial notation.)

N(c1) =

Solution

condition that AMBER appears

AMBER = 5 letter word

26 letter English alphabet - 5 letter word AMBER = 21 letter word

= 21

so there are 21! permutations containing AMBER

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the condition that BERTH appears

BERTH = 5 letter word

26 letter English alphabet - 5 letter word BERTH = 21 letter word

= 21

so there are 21! permutations containing BERTH

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the condition that THINK appears

THINK = 5 letter word

26 letter English alphabet - 5 letter word THINK = 21 letter word

= 21

so there are 21! permutations containing THINK

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If a permutation contains both AMBER and BERTH,

it actually contains AMBERTH = 7 letter word

26 - 7 = 19

so there are 19! permutations containing AMBER and BERTH.

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If a permutation contains both BERTH and THINK

it actually contains BERTHINK = 8 letter word

26 - 8 = 18

so there are 18! permutations containing BERTH and THINK

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If a permutation contains both AMBER and THINK

AMBER THINK = 10 lettrers

26 - 10 = 16

so there are 16! permutations containing AMBER and THINK

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If a permutation contains AMBER,BERTH and THINK

it actually contains AMBERTHINK = 10 letter word

26 - 10 = 16

so there are 16! permutations containing AMBER , BERTH and THINK