Vowels and Consonants

"Darlington" has 10 letters. 3 vowels and 7 consonants. as they need to be in alphabetical order so for vowels there will be only 1 possibility to arrange them in alphabetical order and so will be consonants. Therefore answer is 10c3 or 10c7 which equals 120

$\text{VOWELS}=3$

$\text{CONSONANTS}=7$

All we need is select where vowels or constonants be, this is $\binom{10}{7}=\binom{10}{3}=\boxed{120}$

Here is my approach: _ D_ G _ L _ N _ N _ R _ T

This is the order in which the consonants will appear. We cant change it. We can only choose where to place the vowels obeying the condition that they must be alphabetically arranged.

1. We can put the three vowels AIO in one of the spots so that they come consecutively. This can be done in 8C1 = 8 ways.

2. We can put them in 3 different spots in 8C3 = 56 ways.

3. We can also put them in 2 different spots in 8C2 ways. We can also select which two words to put together in one spot- either AI or IO. Hence, total ways is 8C2 *2 = 56.

Total = 56 + 56 + 8 = 120.