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    how many words with or without meaning can be made from the letters of the word monday

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    How many words, with or without meaning can be made from the letters of the word MONDAY , assuming that no letter is repeated, if.(i) 4 letters are used at a time.(ii) all letters are used at a time(iii) all letters are used but first letter is a vowel?

    Click here👆to get an answer to your question ✍️ How many words, with or without meaning can be made from the letters of the word MONDAY , assuming that no letter is repeated, if.(i) 4 letters are used at a time.(ii) all letters are used at a time(iii) all letters are used but first letter is a vowel?

    How many words, with or without meaning can be made from the letters of the word MONDAY, assuming that no letter is repeated, if.

    Question

    (i) 4 letters are used at a time.(ii) all letters are used at a time(iii) all letters are used but first letter is a vowel?

    Medium Open in App

    Updated on : 2022-09-05

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

    Solution Verified by Toppr

    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 ​ = (6−4)! 6! ​ = 2! 6! ​ = 2! 6×5×4×3×2! ​ =6×5×4×3=360

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

    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 place 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 6×5×4×3×2×1=6!

    Alternative Method:

    Number of words that can be formed by using all the letters of the word MONDAY at a time is the number of permutation of 6 different objects taken 6 at a time, which is

    6 P 6 ​ =6!

    Thus, required number of words that can be formed when all letters are used at a time= 6!=6×5×4×3×2×1=720

    (iii) Total number of letters in the word MONDAY is 6.

    Number of vowels are 2(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, sixth place in 1 way.

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

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    How many words with or without meaning can be made

    How many words with or without meaning can be made from the letters of the word MONDAY assuming that no letter is repeated if i 4 letters are used at a time ii All letters are used at a time iii All l

    How many words, with or without meaning can be made from the letters of the word MONDAY, assuming that no letter is repeated, if,

    (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?

    Answer Verified 159.3k+ views

    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 nCr

    and if we permute r out of n it can be written as

    n P r nPr .

    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 6P4 .

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

    6 P 4 = 6! (6−4)! = 6! 2! = 6×5×4×3×2×1 2×1 =6×5×4 ×3=360

    6P4=6!(6−4)!=6!2!=6×5×4×3×2×12×1=6×5×4×3=360

    As we know n P r nPr is written as n! (n−r)! 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

    6×5×4×3×2×1=6!=720 6×5×4×3×2×1=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! 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

    2×5×4×3×2×1=240 2×5×4×3×2×1=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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    Ex 7.3, 9

    Ex 7.3, 9 How many words, with or without meaning can be made from the letters of the word MONDAY, assuming that no letter is repeated, if 4 letters are used at a time, Total number of alphabets in MONDAY = 6 Hence n = 6 If 4 letters are used at a time, r = 4 Number of different wor

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    Ex 7.3,9 - Chapter 7 Class 11 Permutations and Combinations (Term 2)

    Last updated at Nov. 11, 2020 by Teachoo

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    Transcript

    Ex 7.3, 9 How many words, with or without meaning can be made from the letters of the word MONDAY, assuming that no letter is repeated, if 4 letters are used at a time, Total number of alphabets in MONDAY = 6 Hence n = 6 If 4 letters are used at a time, r = 4 Number of different words = nPr = 6P4 = (6! )/(6 − 4)! = 6!/2! = (6 × 5 × 4 × 3 × 2!)/2! = 6 × 5 × 4 × 3 = 360 Ex7.3, 9 How many words, with or without meaning can be made from the letters of the word MONDAY, assuming that no letter is repeated, if (ii) all letters are used at a time, Total number of alphabets in MONDAY = 6 Hence n = 6 If all letters are used at a time, r = 6 Number of different words = nPr = 6P6 = (6! )/(6 − 6)! = 6!/0! = 6!/1 = 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720 Ex 7.3, 9 How many words, with or without meaning can be made from the letters of the word MONDAY, assuming that no letter is repeated, if (iii) all letters are used but first letter is a vowel? First letter should be a vowel (a, e, i, o, u) Vowel in MONDAY are O and A Assuming first letter O If first letter is O, word will be of the form Number of letters left = 5 n = 5 Number of letters to be used = 5 r = 5 Number of different words = 5P5 = 5!/(5 − 5)! = 5!/0! = 5!/1 = 5! = 5 × 4 × 3 × 2 × 1 = 120 Similarly, If first letter is A , Number of different words = 120 Thus, Required number of words = 120 + 120 = 240

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    Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

    स्रोत : www.teachoo.com

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