The letters of the word EQUATION are arranged in a row. In how many arrangements do all five vowels appear together?

120Correct answerYour answer720Correct answerYour answer2,880Correct answerYour answer40,320Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer C

Correct answer: C

EQUATION has 8 different letters: vowels E, U, A, I, O and consonants Q, T, N.
Treat the 5 vowels as one block. With Q, T and N that makes 4 units: $4! = 24$ orders.
The vowels can be arranged inside the block in $5! = 120$ ways.
Arrangements $= 24 \times 120 = 2880$ .

Worth noting

120 counts only the order inside the block, and 720 multiplies $5!$ by $3!$ , forgetting that the block itself moves among the consonants. 40,320 is $8!$ , with no condition.

Concept: Block method with internal arrangements
Typical timeTypical: 120 s
Your timeYours: —

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Ten questions from Counting and Basic Probability, at your own pace.

The letters of the word EQUATION are arranged in a row. In how many arrangements do all five vowels appear together?

Explanation