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Chapter 1: Problem 9

In how many ways can the letters in WONDERING be arranged with exactly two consecutive vowels?

Short Answer

Expert verified
The letters in the word 'WONDERING' can be arranged with exactly two consecutive vowels in 30240 ways.

Step by step solution

01

Determine the Entity

First, the two vowels need to be arranged. Consider them as one letter. This gives \(3P2 = 6\) arrangements (which comes from the permutation formula for arranging n entities in r places \(nPr\), in this case three vowels arranged in two places).
02

Arrangement of Entities

Secondly, we consider arranging all the entities which includes six consonants and the combined two vowels (considered as one entity). There are 7 distinct entities (6 consonant letters and one group of the two vowels), and they can be arranged in \(7P7 = 5040\) ways.
03

Total Arrangements

Finally, to find total number of arrangements, we multiply the arrangements of the two vowels (step 1) and the total number of arrangements of the entities (step 2). So the total arrangements are \(6*5040 = 30240\) ways.

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Most popular questions from this chapter

A chemistry teacher has seven cartons, each containing 36 test tubes of "unknowns" for a laboratory experiment. The first carton's 36 unknowns comprise four different compounds occurring \(5,12,7,12\) times, respectively. In how many ways can the contents of this carton be distributed among five different chemistry labs?

Four numbers are selected from the following list of numbers: \(-5,-4,-3,-2,-1,1,2,3,4\). (a) In how many ways can the selections be made so that the product of the four numbers is positive and (i) the numbers are distinct? (ii) each number may be selected as many as four times? (iii) each number may be selected at most three times? (b) Answer part (a) with the product of the four numbers negative.

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Find the value of sum after the given Pascal program segment is executed. (Here \(i, j, k\), increment, and sum are integer variables.) $$ \begin{array}{l}\text { inorement }:=0 ; \\ \text { sum : }=0 ; \\ \text { For } i:=1 \text { to } 10 \text { do } \\ \text { For } j:=1 \text { to } 1 \text { do } \\ \text { For } k:=1 \text { to } j \text { do } \\ \text { Begin } \\ \text { inorement : = inorement }+1: \\ \text { sum : sum }+\text { increment }\end{array} $$

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