Porisma

Worked solution

Separate the three vowels of ORANGE

In how many ways can the letters of the word ORANGE be arranged such that all the 3 vowels are separated?

CJC · Lecture · Example 13

Arrange the consonants to create four gaps

The vowels are O,A,E and the consonants R,N,G. Arrange the 3 consonants in 3! ways. There are 4 gaps: before the first consonant, between each neighbouring pair, and after the last consonant.

How did this part go?

Choose distinct gaps and arrange the vowels

Choose 3 of those 4 gaps in (43)\binom43 ways, and arrange O,A,E in the selected gaps in 3! ways. At most one vowel goes in each gap, so no two vowels are adjacent. Total=3!(43) 3!=6×4×6=1443!\binom43\,3!=6\times4\times6=144. This follows the source tip: arrange the objects that need not be separated, then fill the gaps with the objects that must be separated.

How did this part go?

Compare your sketch

Expand diagram

Use the scrollbars or arrow keys to inspect details.

Source gap construction: three consonants, here one illustrative order R,N,G, have four gaps including both ends. Put the three distinct vowels O,A,E into three different gaps. Count three factorial consonant orders, choose three of four gaps, and three factorial vowel orders, giving 144.
Source gap construction: three consonants, here one illustrative order R,N,G, have four gaps including both ends. Put the three distinct vowels O,A,E into three different gaps. Count three factorial consonant orders, choose three of four gaps, and three factorial vowel orders, giving 144.
Source gap construction: three consonants, here one illustrative order R,N,G, have four gaps including both ends. Put the three distinct vowels O,A,E into three different gaps. Count three factorial consonant orders, choose three of four gaps, and three factorial vowel orders, giving 144.

Common mistakes

Both end gaps count. Using different gaps prevents adjacent vowels; arranging the three distinct vowels supplies a separate 3! factor.

Read the hint