JAMB 2017 Mathematics · Question 19

Find the number of ways that the letters of the word EXCELLENCE be arranged

Options

A. $\frac{10!}{2!2!2!}$
B. $\frac{10!}{4!2!}$
C. $\frac{10!}{4!2!2!}$
D. $\frac{10!}{2!2!}$

Answer: C. $\frac{10!}{4!2!2!}$

Explanation written with AI and checked against the answer key

Analysis

To find the number of ways to arrange the letters of a word, we use the formula for permutations of objects with repetitions:

\text{Number of arrangements} = \frac{n!}{n_1! \times n_2! \times \dots \times n_k!}

Where:

$n$ is the total number of letters.
$n_1, n_2, \dots$ are the frequencies of the repeating letters.

Concept

Count the total letters ( $n$ ): The word EXCELLENCE has 10 letters.
Identify repeating letters: • E appears 3 times. • X appears 1 time. • C appears 2 times. • L appears 2 times. • N appears 1 time.

Wait, let's re-count carefully: E - X - C - E - L - L - E - N - C - E

E: 4 times (E, E, E, E)
X: 1 time
C: 2 times (C, C)
L: 2 times (L, L)
N: 1 time

Total letters ( $n$ ) = 10. Frequencies: E=4, C=2, L=2.

Solution

Using the formula:

\text{Arrangements} = \frac{10!}{4! \times 2! \times 2!}

$10!$ represents the total permutations if all letters were unique.
$4!$ accounts for the arrangements of the four 'E's that look identical.
$2!$ accounts for the arrangements of the two 'C's.
$2!$ accounts for the arrangements of the two 'L's.

Comparing this to your options: A. $\frac{10!}{2!2!2!}$ B. $\frac{10!}{4!2!}$ C. $\frac{10!}{4!2!2!}$ D. $\frac{10!}{2!2!}$

The calculation matches Option C.

Tip

When solving permutation problems with repeated letters, always write out the word and tally the count of each letter individually before plugging them into the formula. This prevents missing a letter or miscounting the frequency of a repeating one!

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