Question

26 alphabets have to be arranged in a row such that all the vowels came before Z.

Answer 1

Hint:
Here we will use the concept of permutation to arrange the alphabets. First we will form the conditions of the position of the letter Z. Then we will write the corresponding possible arrangements using the permutation.

Complete Step by Step Solution:
Total number of letters is 26. Therefore,
\[n = 26\]
Now we will form the condition for the position of letter Z and write the corresponding possible arrangements using the permutation.
First, we will assume the letter Z to be at the 6th position in the row.
Therefore the possible arrangements are \[5!20!\].
Similarly, if the letter Z is at the 7th position in the row, then the possible arrangements are \[19!{}^6{P_5}\].
Similarly, if the letter Z is at the 8th position in the row, then the possible arrangements are \[{}^7{P_5} \times 2! \times 18!\].
Similarly, if the letter Z is at the 9th position in the row, then the possible arrangements are \[{}^8{P_5} \times 3! \times 18!\].

Therefore, similar conditions will form for the different positions of letter Z and different numbers of arrangement corresponding to it.

Note:
Here we have to note that we need to apply the concept of permutation not the combination for the arrangement of the letters. As a permutation is defined as the different ways in which a collection of items can be arranged. For example: The different ways in which the numbers 1, 2 and 3 can be grouped together, taken all at a time, are\[123,{\rm{ }}132,{\rm{ }}213{\rm{ }}231,{\rm{ }}312,{\rm{ }}321\].
Combinations may be defined as the various ways in which objects from a set may be selected. For example: The different selections possible from the numbers 1, 2, 3 taking 2 at a time, are \[{\rm{12, 23 \text{ and } 31}}\]