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Nội dung chính Show- How many 3 letters words can be formed?How many 3 letter words with or without meaning can be formed?How many 3 letter words can be formed using Square?How many 3 letter words are there with two letters the same?
Verified
Hint: In this question, we need to consider all the possibilities. Now, calculate the number of 3 letter words that can be formed with all the letters being different using the permutation formula given by [^nP_r]. Then find the number of words having two letters similar using the combinations given by the formula [^nC_r] and then arrange them. Then find the words with all the 3 letters the same and add all these to get the result.
Complete step-by-step
answer:
Now, from the given word SAHARANPUR in the question we have
S, A, A, A, H, R, R, N, P, U
Now, we need to find the 3 letter words that can be formed using these letters
Let us first consider the case that all the 3 letters to be different
Now, we have to arrange the 7 different letters in 3 places
As we already know that arrangement of n things in r places can be done using the permutations given by the formula
[^nP_r=dfracn!left( n-r right)!]
Now,
on comparing with the formula we have
[n=7,r=3]
Now, on substituting the respective values we get,
[Rightarrow ^7P_3]
Now, this can be further written as
[Rightarrow dfrac7!left( 7-3 right)!]
Now, on further simplification we get,
[beginalign
& Rightarrow 7times 6times 5 \
& Rightarrow 210 \
endalign]
Thus, 210 3- letter words can be formed with all letters being different
Now, let us find the
number of 3-letter words having two letters same
Here, the possible two same letters can be either A or R
As we already know that selection can be done using the combinations given by the formula
[^nC_r=dfracn!left( n-r right)!r!]
Now, we can select either of A or R in
[Rightarrow ^2C_1=dfrac2!1!1!]
That means in 2 ways
Now, the next letter can be any of the remaining 6 letters which can be done in
[Rightarrow 6text ways]
Now,
the arrangement of these letters with 2 letters can be done in
[Rightarrow dfrac3!2!]
[Rightarrow 3text ways]
Now, the 3-letter words that can be formed with two letters same is given by
[Rightarrow 2times 6times 3]
Now, on further simplification we get,
[Rightarrow 36]
Thus, there are 36 3- letter words with 2 letters same
Now, we need to find the number of letters with three letters same
Here, A is the only letter that is repeated thrice
So, the word can be AAA which has only 1 way
Now, the total number of 3-letter words that can be formed from the given word are
[Rightarrow 210+36+1]
Now, on simplifying it further we get,
[Rightarrow 247]
Hence, the correct option is (c).
Note:
It is important to note that we need to consider the words having 2 letters the same and three letters the same with the words having all letters different because these all satisfy the given condition. Here, neglecting
any of the cases gives the incorrect option.
It is also to be noted that in the case that we considered having 2 letters to be the same after choosing the letters we need to arrange them because we are finding the words that are possible.
Answer
Verified
Hint: Here, we will find the number of four-letter words that can be formed where the letter R comes most once, that is each letter comes once. Then, we will find the number of four-letter words that can be formed where the letter R comes twice. Finally, we will add the two results to get the number of four-letter words that can be formed by using the letters of the word “HARD WORK”.
Formula Used:
The number of permutations in which a set of [n] objects can be
arranged in [r] places is given by [^nP_r = dfracn!left( n - r right)!], where no object is repeated.
The number of permutations to arrange [n] objects is given by [dfracn!r_1!r_2! ldots r_n!], where an object appears [r_1] times, another object repeats [r_2], and so on.Complete step-by-step answer:
The number of letters in the word ‘HARD WORK are 8, where R comes twice.
The letters are to be arranged in 4 places.
The
number of permutations in which a set of [n] objects can be arranged in [r] places is given by [^nP_r = dfracn!left( n - r right)!], where no object is repeated.
The number of permutations to arrange [n] objects is given by [dfracn!r_1!r_2! ldots r_n!], where an object appears [r_1] times, another object repeats [r_2], and so on.
Thus, we can find the answer using two cases.
Case 1: The letter R is not repeated in the 4 places.
We
have 7 letters to be placed in 4 spaces.
The 7 letters are H, A, R, D, W, O, K.
We observe that no letter is being repeated.
Substituting [n = 7] and [r = 4] in the formula [^nP_r = dfracn!left( n - r right)!], we get
[^7P_4 = dfrac7!left( 7 - 3 right)! = dfrac7!4! = dfrac7 times 6 times 5 times 4 times 3 times 2 times 14 times 3 times 2 times 1 = 840]
Therefore, the number of four-letter words that can be
formed where the letter R comes most once, is 840.
Case 2: The letter R is repeated in the 4 places.
In the 4 places, 2 places will be taken by the two R’s, and the remaining 2 places will be taken by any of the remaining 6 letters.
The number of ways in which this is possible can be found by using combinations.
Therefore, we get
Number of four-letter words where R is repeated (order not important) [ = ^2C_2 times ^6C_2]
Since the order matters in the number of
words we need to find, we will find the order in which the 4 letters (chosen in [^2C_2 times ^6C_2] ways) can be placed in the 4 places, where R is repeated.
This can be found by using the formula [dfracn!r_1!r_2! ldots r_n!].
Thus, the four chosen letters can be ordered in [dfrac4!2!] ways.
Therefore, we get the number of four-letter words where the 7 letters are placed in 4 places, and R is repeated, is given by [^2C_2 times ^6C_2
times dfrac4!2!] ways.
Here, [^2C_2 times ^6C_2] is the number of ways of choosing the letters to be placed within the 4 places, and [dfrac4!2!] is the number of ways in which the chosen 4 letters can be ordered.
Simplifying the expression, we get
Number of four-letter words where the letter R is repeated [ = 1 times dfrac6 times 52 times 1 times dfrac4 times 3 times 2 times 12 times 1 = 180]
Therefore, the number of
four-letter words where the letter R is repeated is 180.
Finally, we will calculate the number of four-letter words that can be formed using the letters of the word “HARD WORK”.
The number of four-letter words that can be formed using the letters of the word “HARD WORK” is the sum of the number of four-letter words that can be formed where the letter R comes most once, and the number of four-letter words that can be formed where the letter R comes twice.
Thus, we get the number of
four-letter words that can be formed using the letters of the word “HARD WORK” is [840 + 180 = 1020] words.
Therefore, 1020 four-letter words can be formed using the letters of the word “HARD WORK”.
Note: We used combinations to get the number of four-letter words where the letter R is repeated (order not important). The number of combinations in which a set of [n] objects can be arranged in [r] places is given by [^nC_r = dfracn!r!left( n - r right)!]. Therefore, the number of ways in which the 2 R’s can be placed in 2 places is [^2C_2], and the number of ways to place the remaining 6 letters in the 2 places is [^6C_2]. By multiplying these, we get the number of ways to place the 8 letters in the 4 places, such that the letter R comes twice, and order of letters does not matter.
How many 3 letters words can be formed?
Thus, there are 24+18 = 42 distinguishable three letter words that can be formed.How many 3 letter words with or without meaning can be formed?
Hence, the number of 3-letter words (with or without meaning) formed by using these letters = 10P3=10×9×8=720.How many 3 letter words can be formed using Square?
The number of 3-letter words is (63)⋅3! =120.How many 3 letter words are there with two letters the same?
To use your method, you need to subtract 2⋅26 -- 26 words which are three letters the same, and 2 because you counted them thrice and you want to count them only once. This gives you 2028−2⋅26=1976, matching the answer from the first method. Tải thêm tài liệu liên quan đến nội dung bài viết Find out the distinct three letter words that can be formed using the word SINGAPORE
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