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(i) 4 letters are used at a time.

(ii) All letters are used at a time.

(iii) All letters are used but the first letter is a vowel?

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Hint: Here we go through by applying the properties of permutation and combination for arranging the letters to form the word. As we know if we select r out of n it can be written as ${}^n{C_r}$ and if we permute r out of n it can be written as ${}^n{P_r}$.

Complete step-by-step answer:

Here in the question the given word is MONDAY. Here we clearly see there are 6 different letters in the word MONDAY.

(i) Number of 4-letter words that can be formed from the letters of the word MONDAY, without repetition of letters, is the number of permutations of 6 different objects taken 4 at a time, which is ${}^6{P_4}$.

Thus, required number of words that can be formed using 4 letters at a time is

${}^6{P_4} = \dfrac{{6!}}{{(6 - 4)!}} = \dfrac{{6!}}{{2!}} = \dfrac{{6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{2 \times 1}} = 6 \times 5 \times 4 \times 3 = 360$ As we know ${}^n{P_r}$ is written as$\dfrac{{n!}}{{(n - r)!}}$.

(ii) There are 6 different letters in the word MONDAY.

When all the letters are used at time then there are six places and the six words in which the first place can be filled in 6 ways.

Second place can be filled by any one of the remaining 5 letters. So, second place can be filled in 5 ways.

Third place can be filled by any one of the remaining 4 letters. So, third place can be filled in 4 ways.

So, on continuing, number of ways of filling fourth place in 3 ways, fifth place in 2 ways, six places in 1 way.

Therefore, the number of words that can be formed using all the letters of the word MONDAY, using each letter exactly once is $6 \times 5 \times 4 \times 3 \times 2 \times 1 = 6! = 720$.

Alternative method for solving this part:

Number of words that can be formed by using all the letters of the word MONDAY at a time is the number of permutations of 6 different objects taken 6 at a time, which is $6!$. (As we know arranging n letters at n place is written by n!).

(iii) Total number of letters in the word MONDAY is 6, Number of vowels are 2 i.e. (O, A)

Six letters word is to be formed.

First letter should be a vowel. So, the rightmost place of the words formed can be filled only in 2 ways.

Since the letters cannot be repeated, the second place can be filled by the remaining 5 letters. So, second place can be done in 5 ways.

Similarly, third place in 4 ways, fourth place in 3 ways, fifth place in 2 ways, and sixth place in 1 way.

Hence, the required number of words that can be formed using four letters of the given word is $2 \times 5 \times 4 \times 3 \times 2 \times 1 = 240$.

Note: Whenever we face such type of question the key concept for solving the question is go through the properties of permutation and combination as we know if there are n places and n numbers are given then first place can filled by n choice then second place can be fill by (n-1) choice and so on. This property is the golden rule for solving such types of questions.

Complete step-by-step answer:

Here in the question the given word is MONDAY. Here we clearly see there are 6 different letters in the word MONDAY.

(i) Number of 4-letter words that can be formed from the letters of the word MONDAY, without repetition of letters, is the number of permutations of 6 different objects taken 4 at a time, which is ${}^6{P_4}$.

Thus, required number of words that can be formed using 4 letters at a time is

${}^6{P_4} = \dfrac{{6!}}{{(6 - 4)!}} = \dfrac{{6!}}{{2!}} = \dfrac{{6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{2 \times 1}} = 6 \times 5 \times 4 \times 3 = 360$ As we know ${}^n{P_r}$ is written as$\dfrac{{n!}}{{(n - r)!}}$.

(ii) There are 6 different letters in the word MONDAY.

When all the letters are used at time then there are six places and the six words in which the first place can be filled in 6 ways.

Second place can be filled by any one of the remaining 5 letters. So, second place can be filled in 5 ways.

Third place can be filled by any one of the remaining 4 letters. So, third place can be filled in 4 ways.

So, on continuing, number of ways of filling fourth place in 3 ways, fifth place in 2 ways, six places in 1 way.

Therefore, the number of words that can be formed using all the letters of the word MONDAY, using each letter exactly once is $6 \times 5 \times 4 \times 3 \times 2 \times 1 = 6! = 720$.

Alternative method for solving this part:

Number of words that can be formed by using all the letters of the word MONDAY at a time is the number of permutations of 6 different objects taken 6 at a time, which is $6!$. (As we know arranging n letters at n place is written by n!).

(iii) Total number of letters in the word MONDAY is 6, Number of vowels are 2 i.e. (O, A)

Six letters word is to be formed.

First letter should be a vowel. So, the rightmost place of the words formed can be filled only in 2 ways.

Since the letters cannot be repeated, the second place can be filled by the remaining 5 letters. So, second place can be done in 5 ways.

Similarly, third place in 4 ways, fourth place in 3 ways, fifth place in 2 ways, and sixth place in 1 way.

Hence, the required number of words that can be formed using four letters of the given word is $2 \times 5 \times 4 \times 3 \times 2 \times 1 = 240$.

Note: Whenever we face such type of question the key concept for solving the question is go through the properties of permutation and combination as we know if there are n places and n numbers are given then first place can filled by n choice then second place can be fill by (n-1) choice and so on. This property is the golden rule for solving such types of questions.

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