Arranging all the Letters of a Word where all the letters are different

No. Problems & Solutions [Click Hide/Show to display the solutions below the question]
04.
Define the Event and identify the number of favourable choices in the following cases
a) The letters of the word SLAUGHTER are arranged in a row at random. Find the probability that the vowels may be in the odd places.
b) The letters of the word "RIGHTEOUSLY" are arranged in a row at random. Find the probability that the even places are filled with consonants.
c) The letters of the word “PENTAGON” are arranged in a row at random. Find the probability that there are exactly two letters between E and A is
d) Each of the letters A, B, E, L, T are written on a separate card. If all the cards are arranged in a row in all possible ways, the probability of the forming the word TABLE is

Solution (a)  
 
Number of Letters/Characters in the word "SLAUGHTER" = 9 {S, L, A, U, G, H, T, E, R}
⇒ nL = 9

In the experiment of testing for the number of words that can be formed using the letters of the word "SLAUGHTER"

Total No. of Possible Choices = Number of words that can be formed using the 9 letters of the word "SLAUGHTER"
⇒ n = 9P9 (Or) 9!
Let "A" be the event of the word formed with all the vowels occupying odd places

For Event "A"

No. of vowels (a) = 3 {A, U, E}
No. of other letters (Consonants) = 6 {S, L, G, H, T, R}
No. of odd places = 5 {_, X, _, X, _, X, _, X, _}

Number of Favourable/Favorable Choices = The no. of words that can be formed using the letters of the word
"SLAUGHTER" such that the vowels occupy odd places.
mA = (No. of ways in which the 3 vowels can be arranged in the 5 odd places)
× (No. of ways in which the remaining 6 letters can be arranged
      in the remaining 6 places)
= 5P3 × 6P6
= 5 × 4 × 3 × 6!
= 60 × 720
= 43,200

Solution (b)  
 
Number of Letters/Characters in the word "RIGHTEOUSLY" = 11{R, I, G, H, T, E, O, U, S, L, Y}
⇒ nL = 11

In the experiment of testing for the number of words that can be formed using the letters of the word "RIGHTEOUSLY"

Total No. of Possible Choices = Number of words that can be formed using the 9 letters of the word "RIGHTEOUSLY"
⇒ n = 11P11 (Or) 11!
Let "H" be the event of forming the words such that the consonants occupy even places

For Event "H"

No. of Consonants = 7 {R, G, H, T, S, L, Y}
No. of other letters (Vowels) = 4 {I, E, O, U}
No. of even places = 5 {X, _, X, _, X, _, X, _, X, _, X}

Number of Favourable/Favorable Choices = The no. of words that can be formed using the letters of the word
"RIGHTEOUSLY" such that the even places are filled with consonants
mH = (No. of ways in which the 5 even places can be filled with the 7
      consonants)
× (No. of ways in which the remaining 6 places can be filled with the
      remaining 6 letters)
= 7P5 × 6P6
= 7 × 6 × 5 × 4 × 3 × 6!
= 2,520 × 720
= 18,14,400

Solution (c)  
 
Number of Letters/Characters in the word "PENTAGON" = 8 {P, E, N, T, A, G, O, N}
⇒ nL = 8

In the experiment of testing for the number of words that can be formed using the letters of the word "PENTAGON"

Total No. of Possible Choices = Number of words that can be formed using the 8 letters of the word "PENTAGON"
⇒ n = 8P8 (Or) 8!
Let "T" be the event of forming the words such that there are exactly two letters between E and A

For Event "T"

Event "T" can be accomplished in five alternative ways, with "E" and "A" occupying

  1. The First and the Fourth places {_, X, X, _, X, X, X, X} → (Ta)
  2. The Second and the Fifth places {X, _, X, X, _, X, X, X} → (Tb)
  3. The Third and the Sixth places {X, X, _, X, X, _, X, X} → (Tc)
  4. The Fourth and the Seventh places {X, X, X, _, X, X, _, X} → (Td)
  5. The Fifth and the Eighth places {X, X, X, X, _, X, X, _ } → (Te)

• 1st Alternative (Ta)

Number of Favourable/Favorable Choices = The no. of words that can be formed using the letters of the word
"PENTAGON" filling E and A in the first and third places.
mTa = (No. of ways in which the 6 letters other than "E" and "A" can
      be filled in the "6" places)
× (No. of ways in which "E" and "A" can be inter arranged in the
      two places designated to them)
= 6P6 × 2P2
= 6! × 2!
= 720 × 2
= 1,440

Similarly, each of the alternative events can be accomplished in 1,440 ways.

Thus, mTa = mTb = mTc = mTd = mTe = 1,440

Total Number of Favourable/Favorable Choices = Sum of the number of ways in which each alternative event
    can be accomplished.
mT = mTa + mTb + mTc + mTd + mTe
= 1,440 + 1,440 + 1,440 + 1,440 + 1,440
= 7,200

Solution (d)  
 
Number of Letters/Characters given = 5 {A, B, E, L, T}
⇒ nL = 5

In the experiment of testing for the number of words that can be formed using the given letters

Total No. of Possible Choices = Number of words that can be formed using the 5 letters
⇒ n = 5P5 (Or) 5!
= 120
Let "B" be the event of forming the words such that the word formed is TABLE

For Event "B"

For forming the word "TABLE" each letter has to be fixed in a particular place.

Number of Favourable/Favorable Choices = (No. of ways in which the Letter "T" can be filled in the First place)
× (No. of ways in which the Letter "A" can be filled in the Second place)
× (No. of ways in which the Letter "B" can be filled in the Third place)
× (No. of ways in which the Letter "L" can be filled in the Fourth place)
× (No. of ways in which the Letter "E" can be filled in the Fifth place)
= 1P1 × 1P1 × 1P1 × 1P1 × 1P1
= 1 × 1 × 1 × 1 × 1
= 1

Author Credit : The Edifier  

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