**Introduction**

This question is from a source that does
not credit the original source. In the
original question was poorly worded. It
did not have the words “

*the letters of the word*”. Instead of “*each*vowel must be separated”, it said “a vowel must be separated”, which is might mean there is just one such instance. This is ambiguous. I have taken the liberty to rephrase some parts of the question to make its meaning clearer. In this article, I shall discuss only part (ii) of this question.**Stage 1: Understanding the question**

What is the given in the problem? Can you organise the information?

Though not absolutely necessary,
it is helpful to draw a diagram that separate the letters into vowels and
consonants and write the stack up the same letters in columns.
There are 5 vowels (of which O is repeated) and
6 consonants (of which N
and S are repeated).

Can you explain the problem in
your own words?

The letters of the word ‘CONNOISSEUR’ are re-arranged, which means that
all the 11 letters are used. Each vowel
must be separated from another with exactly one consonant, which means that the
letters must

*contain*the pattern “v c v c v c v c v” (where v = vowel, c = consonant). Important: Note that the question does not say that the first letter must be a consonant.**Stage 2: Planning**

Have you seen a similar problem
before?

Yes, but this looks a bit more challenging. There are more possibilities as first letter need
not be a consonant.

What heuristics can you try?

· Solve part of the problem

· Split the problem into smaller problems

What topic-specific tactics can
you try?

The “v c v c v c v c v” pattern can be treated as a group (Grouping Method). Since there are six consonants, there are two
more “c”s (consonants) in the full pattern.
This looks like a problem that can use the Insertion Method.

**Stage 3: Execution**

The number of ways to insert the
group =

^{3}C_{1}= 3
[these are the patterns “ccvcvcvcvcv”, “cvcvcvcvcvc” and “vcvcvcvcvcc” ]

For each pattern,

the vowels can be arranged in

^{5!}/_{2!}ways (division because there are 2 ‘O’)
the consonants can be arranged in

^{6!}/_{2!}_{2!}ways (division because of 2 ‘N’ and 2 ‘S’)
Hence the total number of ways is

**Stage 4: Evaluation**

Is the answer correct?

Yes, the answer is correct.

**Stage 5: Reflection**

What have we learned by solving
this problem?

We have learned once again that heuristics and metacognition are useful
in solving mathematical problems. Specifically,
we have used the following heuristics:-

· Drawing a diagram

· Solve part of the
problem

· Split the problem
into smaller problems

We have also used the following techniques that are useful for
combinatorical problems:-

· Grouping Method

· Insertion Method

· Division Method (for
dealing with repeated letters)

It is also important to understand the problem correctly and not make unfounded
assumptions. If the wording is not
clear, you may want to rephrase it.