## EDUCATION

In how many arrangements of the letters in the nine-letter word EDUCATION are the vowels in alphabetical order?
Source: mathcontest.olemiss.edu 11/15/2010

SOLUTION
The five vowels in alphabetical order are aeiou. An example of an arrangement is $\mathit{ae}-\mathit{i}-\mathit{o}--\mathit{u}$ where the four dashes are placeholders for the four consonants $\mathit{cdnt}$. And in an arrangement of the vowels, there are $4\times3\times2\times1= 24$ ways of placing the four consonants. Thus, to find the answer to our problem, all we have to do is find the total number of arrangements in alphabetical order of the vowels and multiply it by twenty four. The result will be the answer.

STEP 1 How many ways can we break up five vowels aeiou?
To find all the possible arrangements of the five vowels, we need to ask ourselves in how many ways can we partition the number $5$? We can partition $5$ into one, two, three, four, or five parts.

1Partition $5$ into one part
This is a special case. Partition $5$ into one part really means leaving it alone. It means all the five vowels are bunched together, for example, $-\mathit{aeiou}---$.

2Partition $5$ into two parts
There are four ways:

$1+4$ corresponding to $\mathit{a}|\mathit{eiou}$. For example, $\mathit{a}-\mathit{eiou}---$.

$4+1$ corresponding to $\mathit{aeio}|\mathit{u}$. For example, $---\mathit{aeio}-\mathit{u}$.

$2+3$ corresponding to $\mathit{ae}|\mathit{iou}$. For example, $\mathit{ae}----\mathit{iou}$.

$3+2$ corresponding to $\mathit{aei}|\mathit{ou}$. For example, $-\mathit{aei}-\mathit{ou}--$.

3Partition $5$ into three parts
There are six ways:

$1+1+3$corresponding to $\mathit{a}|\mathit{e}|\mathit{iou}$. For example, $\mathit{a}-\mathit{e}-\mathit{iou}--$.

$1+2+2$ corresponding to $\mathit{a}|\mathit{ei}|\mathit{ou}$. For example, $-\mathit{a}-\mathit{ei}-\mathit{ou}-$.

$1+3+1$ corresponding to $\mathit{a}|\mathit{eio}|\mathit{u}$. For example, $\mathit{a}--\mathit{eio}--\mathit{u}$.

$3+1+1$ corresponding to $\mathit{aei}|\mathit{o}|\mathit{u}$. For example, $\mathit{aei}--\mathit{o}--\mathit{u}$.

$2+2+1$ corresponding to $\mathit{ae}|\mathit{io}|\mathit{u}$. For example, $\mathit{ae}-\mathit{io}---\mathit{u}$.

$2+1+2$ corresponding to $\mathit{ae}|\mathit{i}|\mathit{ou}$. For example, $\mathit{ae}---\mathit{i}-\mathit{ou}$.

4Partition $5$ into four parts
There are four ways:

$1+1+1+2$ corresponding to $\mathit{a}|\mathit{e}|\mathit{i}|\mathit{ou}$. For example, $-\mathit{a}-\mathit{e}-\mathit{i}-\mathit{ou}$.

$2+1+1+1$ corresponding to $\mathit{ae}|\mathit{i}|\mathit{o}|\mathit{u}$. For example, $\mathit{ae}-\mathit{i}-\mathit{o}-\mathit{u}-$.

$1+1+2+1$ corresponding to $\mathit{a}|\mathit{e}|\mathit{io}|\mathit{u}$. For example, $\mathit{a}-\mathit{e}-\mathit{io}-\mathit{u}-$.

$1+2+1+1$ corresponding to $\mathit{a}|\mathit{ei}|\mathit{o}|\mathit{u}$. For example, $\mathit{a}--\mathit{ei}-\mathit{o}-\mathit{u}$.

5Partition $5$ into five parts
There is only one way:

$1+1+1+1+1$corresponding to $\mathit{a}|\mathit{e}|\mathit{i}|\mathit{o}|\mathit{u}$. For example, $\mathit{a}-\mathit{e}-\mathit{i}-\mathit{o}-\mathit{u}$.

STEP 2 Now, we examine each case in details.

Case 1 All five vowels in one part.
There are five arrangements:

$\mathit{aeiou}----$

$-\mathit{aeiou}---$

$--\mathit{aeiou}--$

$---\mathit{aeiou}-$

$----\mathit{aeiou}$

Case 2 Partition the vowels into two parts.
2.1 $1+4$ corresponding to $\mathit{a}|\mathit{eiou}$.

There are ten arrangements:

$\mathit{a}-\mathit{eiou}---$

$\mathit{a}--\mathit{eiou}--$

$\mathit{a}---\mathit{eiou}-$

$\mathit{a}----\mathit{eiou}$

$-\mathit{a}-\mathit{eiou}--$

$-\mathit{a}--\mathit{eiou}-$

$-\mathit{a}---\mathit{eiou}$

$--\mathit{a}-\mathit{eiou}-$

$--\mathit{a}--\mathit{eiou}$

$---\mathit{a}-\mathit{eiou}$

2.2 For the remaining three partitions in this category,

(1) $4+1$ corresponding to $\mathit{aeio}|\mathit{u}$,
(2) $2+3$ corresponding to $\mathit{ae}|\mathit{iou}$,
(3) $3+2$ corresponding to $\mathit{aei}|\mathit{ou}$,

there are similarly ten arrangements per each partition. You can try arranging them yourself for your satisfaction.

Thus, when the vowels are partitioned into two parts, there are a total of $10\times4=40$ arrangements.

Case 3 Partition the vowels into three parts.
3.1 $1+1+3$ corresponding to $\mathit{a}|\mathit{e}|\mathit{iou}$

There are ten arrangements:

$\mathit{a}-\mathit{e}-\mathit{iou}--$

$\mathit{a}-\mathit{e}--\mathit{iou}-$

$\mathit{a}-\mathit{e}---\mathit{iou}$

$\mathit{a}--\mathit{e}-\mathit{iou}-$

$\mathit{a}--\mathit{e}--\mathit{iou}$

$\mathit{a}---\mathit{e}-\mathit{iou}$

$-\mathit{a}-\mathit{e}-\mathit{iou}-$

$-\mathit{a}-\mathit{e}--\mathit{iou}$

$-\mathit{a}--\mathit{e}-\mathit{iou}$

$--\mathit{a}-\mathit{e}-\mathit{iou}$

3.2 For the remaining five partitions in this category,

(1) $1+2+2$ corresponding to $\mathit{a}|\mathit{ei}|\mathit{ou}$,
(2) $1+3+1$ corresponding to $\mathit{a}|\mathit{eio}|\mathit{u}$,
(3) $3+1+1$ corresponding to $\mathit{aei}|\mathit{o}|\mathit{u}$,
(4) $2+2+1$ corresponding to $\mathit{ae}|\mathit{io}|\mathit{u}$,
(5) $2+1+2$ corresponding to $\mathit{ae}|\mathit{i}|\mathit{ou}$,

there are similarly ten arrangements per each partition.

Thus, when the vowels are partitioned into three parts, there are a total of $10\times6=60$ arrangements.

Case 4 Partition the vowels into four parts.
4.1 $1+1+1+2$ corresponding to $\mathit{a}|\mathit{e}|\mathit{i}|\mathit{ou}$

There are five arrangements:

$\mathit{a}-\mathit{e}-\mathit{i}-\mathit{ou}-$

$\mathit{a}-\mathit{e}-\mathit{i}--\mathit{ou}$

$\mathit{a}-\mathit{e}--\mathit{i}-\mathit{ou}$

$\mathit{a}--\mathit{e}-\mathit{i}-\mathit{ou}$

$-\mathit{a}-\mathit{e}-\mathit{i}-\mathit{ou}$

4.2 For the remaining three partitions in this category,

(1) $2+1+1+1$ corresponding to $\mathit{ae}|\mathit{i}|\mathit{o}|\mathit{u}$,
(2) $1+1+2+1$ corresponding to $\mathit{a}|\mathit{e}|\mathit{io}|\mathit{u}$,
(3) $1+2+1+1$ corresponding to $\mathit{a}|\mathit{ei}|\mathit{o}|\mathit{u}$,

there are similarly five arrangements per each partition.

Thus, when the vowels are partitioned into four parts, there are a total of $5\times4=20$ arrangements.

Case 5 Partition the vowels into five parts.
$1+1+1+1+1$ corresponding to $\mathit{a}|\mathit{e}|\mathit{i}|\mathit{o}|\mathit{u}$

There is one arrangement:
$\mathit{a}-\mathit{e}-\mathit{i}-\mathit{o}-\mathit{u}$

STEP 3 Calculate the total number of arrangements
The following table summarizes the results of our investigation.

 Number of parts in vowels Number of arrangements 1 5 2 40 3 60 4 20 5 1 Total = 126 arrangements

For each of the above arrangements of the vowels, there are $4\times3\times2\times1=24$ ways to choose the consonants. Therefore, in $126\times24=3024$ arrangements of the letters in the nine-letter word EDUCATION the vowels are in alphabetical order.

Alternative solution:
Finding the number of arrangements of the 9 letters where the vowels are in alphabetical order can be modeled by three tasks:
(1) select 4 positions to place the consonants $\mathit {c,d,n,t}$
(2) select an ordering of the 4 consonants
(3) fill in the remaining 5 positions with $\mathit{a,e,i,o,u}$ in alphabetical order
Task (1) can be done in $_9C_4$ ways; task (2) can be done in $_4P_4$ ways; task (3) can be done in 1 way.
Thus, the number of arrangements where the vowels are in order equals
$_9C_4 \times\, _4P_4=\frac{9!}{4!5!}\times 4!$

$=\frac{9!}{5!}$
$= \frac{9\times 8\times 7\times 6\times 5!}{5!}$
$= 9\times 8\times 7\times 6$
$=3024$