Multiple choice

How many 6-letter words with distinct letters in each can be formed using the letters of the word EDUCATION? How many of these begin with I

  1. $\displaystyle ^{9}P_{6'}$ $\displaystyle ^{8}P_{5}$
  2. $\displaystyle ^{9}P_{6'}$ $\displaystyle ^{9}P_{5}$
  3. $\displaystyle ^{8}P_{6'}$ $\displaystyle ^{8}P_{5}$
  4. $\displaystyle ^{8}P_{6'}$ $\displaystyle ^{8}P_{4}$
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A Correct answer
Explanation

EDUCATION contains 9 distinct letters, so the number of six-letter arrangements is 9P6. If the word begins with I, the remaining five positions can be filled from the other 8 letters in 8P5 ways.

AI explanation

The word EDUCATION contains 9 distinct letters. The number of 6-letter words with distinct letters that can be formed is found using the permutation formula 9P6. For the second part, if a word must begin with I, we fix I in the first position and must choose and arrange 5 letters from the remaining 8 distinct letters. The number of ways to do this is 8P5.

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