Review How many different four letter words can be formed with the letters of word PACIFIC
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Nội dung chính Show- How many 4 letter words can be formed using?How many 4 letter words can be formed from the letters of the word combination?What is a four letter word p?How many different four
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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 4 letter words can be formed using?
Hence, The total number of four-letter words that can be formed is 270. Therefore, option D. is the correct answer.How many 4 letter words can be formed from the letters of the word combination?
The number of 4 letter words that can be formed using the letters of the word COMBINATION is. a. 2436.What is a four letter word p?
The Four Letter Words that Start with P are peak, poke, pest, pain, pant, pour, pond, poem, pond, pray, prey, park, pass, path, pale, pack, peer, pear, pony, pair. etc.How many different four
Therefore, the total number of possible cases by adding both the cases together. Hence, if the four letter words (need not to be meaningful) are to be formed using the letter from the word “MEDITERRANEAN” such that the first letter is R and the fourth letter is E, then the total number of all such words is 59. Tải thêm tài liệu liên quan đến nội dung bài viết How many different four letter words can be formed with the letters of word PACIFIC
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