1-step method for separating objects in counting questions

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By Brent Hanneson - December 10, 2020

When it comes to counting questions that ask us to arrange a set of objects such that certain objects cannot be adjacent, the traditional approach is to apply the following property:

# of ways to follow restriction = (# of ways to ignore the restriction) - (# of ways to break the restriction)

So, for example, if we must seat A, B, C, D and E in a row of 5 chairs, and A and B are cannot sit together, we'd first IGNORE the restriction and count the number of ways to arrange A, B, C, D and E (120 ways).

Then, we'd count the total number of ways to BREAK the restriction (there are 48 arrangements in which A and B are seated together.

So, the answer = 120 – 48 = 72

As you can see, the traditional approach requires us to complete two steps:

1) Count the number of ways to ignore the restriction

2) Count the number of ways to break the restriction

Consider the following question:

In how many ways can the letters of the word MANIFOLD be arranged so that the vowels are separated?

A. 14,400

B. 18,000

C. 22,200

D. 24,000

E. 36,000

If you try solving this question using the traditional approach, you'll quickly run into problems when trying to count the number of ways to break the restriction that vowels can’t be adjacent.

Instead, we can solve the question as follows:

Stage 1: Arrange the 5 consonants (M, N, F, L and D)

Since n unique objects can be arranged in n! ways, we can arrange the 5 consonants in 5! ways. In other words, we can complete stage 1 in 120 ways.

Key step: For each arrangement of the 5 consonants, add a space on each side of each letter. So, for example, if we add spaces to the arrangement MDLFN, we get: _M_D_L_F_N_

Each of the six spaces represents a possible location for each of the 3 vowels. Notice that this configuration ensures that no vowels can be adjacent.

Stage 2: Choose a space to place the vowel A.

There are 6 available spaces, so we can complete stage 2 in 6 ways.

Stage 3: Choose a space to place the vowel I.

Since there are 5 available spaces remaining, we can complete stage 3 in 5 ways.

Stage 4: Choose a space to place the vowel O.

Since there are 4 available spaces remaining, we can complete stage 4 in 4 ways.

At this point, we'll throw away the remaining spaces, leaving an arrangement with all 8 letters.

By the Fundamental Counting Principle (FCP), the number of ways to complete all 4 stages (and thus place all 8 letters) = (120)(6)(5)(4)  = 14,400 ways.