n_L : Number of letters with which the words are to be formed

n_P : Number of places to fill the letters i.e. the number of letters in the word to be formed

n_P ≤ n_L

Since the maximum length of the word that can be formed is limited to the number of letters available

Number of n_P lettered words that can be formed using n_L letters where all the letters are different

⇒ Number of words that can be formed using n_L letters taking n_P letters at a time

⇒ Number of permutations of n_L items taking n_P at a time

Explanation

• Total Event (E)

  Arranging the n_L letters in n_P places

The total event can be divided into the sub-events of placing each letter in a place starting from the first.

• 1^st sub-event (SE₁)

  Placing a letter in the First place

  The 1^st place can be filled with any one of the available n_L letters

  ⇒ SE₁ can be accomplished in n_L ways

  ⇒ n_SE1 = n_L

• 2^nd sub-event (SE₂)

  Placing a letter in the Second place

  The 2^nd place can be filled with any one of the remaining n_L − 1 letters

  ⇒ SE₂ can be accomplished in n_L − 1 ways

  ⇒ n_SE2 = n_L − 1

• 3^rd sub-event (SE₃)

  Placing a letter in the Third place

  The 3^rd place can be filled with any one of the remaining n_L − 2 letters

  ⇒ SE₃ can be accomplished in n_L − 2 ways

  ⇒ n_SE3 = n_L − 2

• ...
• ...
• n_P^th sub-event (SE_nP)

  Placing a letter in the n_P^th place

  The n_P^th place can be filled with any one of the n_L − (n_P − 1) letters remaining after filling the first n_P − 1 places

  ⇒ SE₃ can be accomplished in n_L − (n_P − 1) ways

  ⇒ n_SEnp = n_L − n_P + 1

By the fundamental counting principle of multiplication,

Number of ways in which the n_L letters can be filled in the n_P places

Example

The number of 5 letter words that can be formed with the letters of the word subdermatoglyphic

Where all letters are used

n_L = n_P

Number of ways in which the n_L letters can be filled in the n_P places

Example

The number of words that can be formed with the letters of the word Algebra

n_L : Number of letters with which the words are to be formed

n_P : Number of places to fill the letters i.e. the number of letters in the word to be formed

n_P ≤ n_L

Since the maximum length of the word that can be formed is limited to the number of letters available

n_FL : Number of letters to be fixed each in its own place

After fixing n_FL letters each in its own place

Number of Letters remaining

= Total number of letters − Number of letters to be fixed

⇒ n_RL = n_L − n_FL

Number of Places remaining to be filled

= Total number of places − Number of letters to be fixed

⇒ n_RP = n_P − n_FL

Number of n_P letter words that can be formed using n_L letters by fixing n_FL letters each in its own place is given by

^n_RLP_nRP

Explanation

• Total Event (E)

  Arranging the n_L letters in n_P spaces

Assume that the total event is divided into two sub-events.

• 1^st sub-event (SE₁)

  Arranging the n_FL letters each in its own place

  Number of ways in which this event can be accomplished

  = 1

  Since each letter can be placed in a specific place only there is only one way this can be done whatever may be the number of letters being fixed

  ⇒ n_SE1 = 1

• 2^nd sub-event (SE₂)

  Filling the remaining n_RP places with the remaining n_RL letters

  Number of ways in which this event can be accomplished

  	=	Number of ways in which nRP places can be filled with the nRL letters
  Or	=	Number of permutations or arrangements of nRL items taking nRP items at a time

  ⇒ nSE2

  =

  nRLPnRP

The number of ways in which the n_L letters can be filled in the n_P places with n_FL letters fixed each in its own place

Fundamental Counting principle of Multiplication

Example

Number of 7 letter words that can be formed with the letters of the word incomputably such that T occupies the middle place

Where all letters are used

n_L = n_P

⇒ n_L − n_FL = n_P − n_FL

⇒ n_RL = n_RP

Number of n_P lettered words that can be formed with the n_L letters, fixing n_FP letters each in its own place

Example

The number of words that can be formed with the letters of the word Thursday such that T occupies the first place and Y occupies the last place

n_L : Number of letters with which the words are to be formed

n_P : Number of places to fill the letters i.e. the number of letters in the word to be formed

n_P ≤ n_L

The maximum length of the word that can be formed is limited to the number of letters available.

n_SL : Number of letters specified to be filled in the specified places

n_SP : Number of places specified to be filled with the specified letters

n_FP : Number of places filled

If n_SL ≠ n_SP,

Number of places filled would be the lesser of n_SL and n_SP.

If the specified letters are lesser only that many letters can be used up for filling and if the specified places are lesser only that many places can be filled.

⇒ n_FP = smaller of n_SL and n_SP

Where n_SL = n_SP,

n_FP = n_SP = n_SL

After fixing n_FL letters each in its own place

Number of Letters remaining

= Total number of letters − Number of letters to be fixed

⇒ n_RL = n_L − n_FL

Number of Places remaining to be filled

= Total number of places − Number of letters to be fixed

⇒ n_RP = n_P − n_FL

n_RP ≤ n_RL

The places remaining to be filled cannot exceed the letters remaining to be used

Number of n_P letter words that can be formed using n_L letters by fixing n_FL letters each in its own place is given by

n_RLPn_RP

Explanation

• Total Event (E)

  Arranging the n_L letters in n_L spaces

Assume that the total event is divided into two sub-events.

• 1^st sub-event (SE₁)

  Arranging the n_SL specified letters in the n_SP specified places

  Number of ways in which this event can be accomplished

  	=	Number of ways in which the nSP specified places can be filled with the nSL specified letters
  Or	=	Number of permutations or arrangements with greater of nSP and nSL items taking the lesser of them at a time

  ⇒ nSE1

  =

  aPb Number of permutations of a items taking b at a time Where a = larger of nSL and nSP and b the other If nSL = nSP, a = b = nSL = nSP

• 2^nd sub-event (SE₂)

  Arranging the remaining letters in the remaining places.

  Number of ways in which this event can be accomplished

  	=	Number of ways in which nRP places can be filled with the nRL letters
  Or	=	Number of permutations or arrangements of nRL items taking nRP at a time

  ⇒ nSE2

  =

  nRLPnRP

The number of words that can be formed

Examples

The number of words that can be formed with the letters of the word Equation such that the even spaces are occupied by vowels

The number of 8 letter words that can be formed with the letters of the word warehousing such that odd positions have only consonants.

n_L : Number of letters with which the words are to be formed

n_P : Number of places to fill the letters i.e. the number of letters in the word to be formed

n_P ≤ n_L

The maximum length of the word that can be formed is limited to the number of letters available.

n_GL1 : Number of letters in the first group

n_GL2 : Number of letters in the second group

n_G : Number of groups of letters

• Total Event (E)

  Arranging the n_L letters in n_P spaces

Assume that the total event is divided into n_G + 1 sub-events.

• 1^st sub-event (SE₁)

  Arranging the n_GL letters in as many places,

  Number of ways in which this event can be accomplished

  	=	Number of ways in which nGL places can be filled with as many letters
  Or	=	Number of permutations or arrangements with nGL items taking all at a time

  ⇒ nSE1	=	nGLPnGL
  	=	nGL!

• 2^nd sub-event (SE₂)

  Arranging the letters in the first group among themselves.

  Number of ways in which this event can be accomplished

  	=	Number of ways in which nGL1 places can be filled with as many letters
  Or	=	Number of permutations or arrangements with nGL1 items taking all at a time

  ⇒ nSE2	=	nGL1PnGL1
  	=	nGL1!

• 3^rd sub-event (SE₃)

  Arranging the letters in the second group among themselves.

  Number of ways in which this event can be accomplished

  	=	Number of ways in which nGL2 places can be filled with as many letters
  Or	=	Number of permutations or arrangements with nGL2 items taking all at a time

  ⇒ nSE3	=	nGL2PnGL2
  	=	nGL2!

• ...
• ...

The number of words that can be formed

Examples

The number of words that can be formed with the letters of the word Victory such that all the vowels come together

The number of words that can be formed with the letters of the word Daughters such that all the vowels come together and DGHT stay together.

n_L : Number of letters with which the words are to be formed

n_P : Number of places to fill the letters i.e. the number of letters in the word to be formed

n_P ≤ n_L

The maximum length of the word that can be formed is limited to the number of letters available.

n_DL : Number of letters to stay divided/separate

Places to arrange letters to stay separate

Using

• one letter we can keep two letters separate.

  OL

• two letters we can keep three letters separate

  OL

  OL

• ...
• n letters we can keep n + 1 letters separate

Thus in finding the number of places available to place the letters to stay separate, consider an arrangement of other letters with places on either side

The places on either side of the other letters are the places where the letters to be divided/separated can appear to ensure that they do not come together.

n_DP ≥ n_DL

To be able to keep n_DL letters separate we need at least as many spaces (n_DP) to fill them up.

If n_DP < n_DL, then it would not be possible to ensure that the n_DL letters stay separate.

• n_DP < n_DL

  Eg : Arrange the letters of the word UTOPIA such that no two vowels come together.

  In the word UTOPIA

  Number of letters

  = 6

  {U, T, O, P, I, A}

  ⇒ n_L = 6

  Number of Letters to stay separate

  ⇒ Number of Vowels

  = 4

  {U, O, I, A}

  ⇒ n_DL = 4

  Number of other letters

  = Total number of letters − Number of letters to stay separate

  ⇒ nOL	=	nL − nDL
  	=	6 − 4
  	=	2

  Number of places to place the letters to stay separate

  = Number of other letters + 1

  ⇒ nDP	=	nOL + 1
  	=	4 + 1
  	=	5

  Since n_DP < n_DL, it would not be possible to arrange the letters in such a way that the vowels do not come together.

  U

  T

  O

  P

  I

  A

  There are only two other letters, T and P. They can separate a maximum of 3 vowels. The fourth vowel would have to come beside another vowel.

• n_DP = n_DL

  When n_DP = n_DL, the specified letters would stay separate only if the word starts as well as ends with one of the letters to stay separate.

  Eg : Arrange the letters of the word Anxious such that no two vowels come together.

  In the word Fortune

  Number of letters

  = 7

  {A, N, X, I, O, U, S}

  ⇒ n_L = 7

  Number of Letters to stay separate

  ⇒ Number of Vowels

  = 4

  {A, I, O, U}

  ⇒ n_DL = 4

  Number of other letters

  = Total number of letters − Number of letters to stay separate

  ⇒ nOL	=	nL − nDL
  	=	7 − 4
  	=	3

  Number of places to place the letters to stay separate

  = Number of other letters + 1

  ⇒ nDP	=	nOL + 1
  	=	3 + 1
  	=	4

  Since n_DP ≥ n_DL, it would be possible to arrange the letters in such a way that the vowels do not come together.

  A

  N

  I

  X

  O

  S

  U

  We can neither start nor end the word with one of the other letters.

• n_OL = n_DL or n_DP = n_DL + 1

  When n_OL = n_DL, ensuring that the letters to stay separate are arranged in the places specified for them would result in the other letters also staying separate.

  Eg : Arrange the letters of the word NATURE such that no two vowels come together.

  In the word NATURE

  Number of letters

  = 6

  {N, A, T, U, R, E}

  ⇒ n_L = 6

  Number of Letters to stay separate

  ⇒ Number of Vowels

  = 3

  {A, U, E}

  ⇒ n_DL = 3

  Number of other letters

  = Total number of letters − Number of letters to stay separate

  ⇒ nOL	=	nL − nDL
  	=	6 − 3
  	=	3

  Number of places to place the letters to stay separate

  = Number of other letters + 1

  ⇒ nDP	=	nOL + 1
  	=	2 + 1
  	=	3

  Since n_DP ≥ n_DL, it would be possible to arrange the letters in such a way that the vowels do not come together.

  A

  N

  U

  R

  E

  T

  N

  U

  R

  E

  T

  A

  The letters would stay separated whether the word starts with a letter from the letters to stay separate or other letters.

• Total Event (E)

  Arranging the n_L letters in n_P spaces

Assume that the total event is divided into two sub-events.

• 1^st sub-event (SE₁)

  Arranging the n_OL letters in n_OL places.

  Number of ways in which this event can be accomplished

  	=	Number of ways in which nOL places can be filled with as many letters
  Or	=	Number of permutations or arrangements with nOL items taking all at a time

  ⇒ nSE1	=	nOLPnOL
  	=	nOL!

• 2^nd sub-event (SE₂)

  Arranging the n_DL letters in the n_DP places that would ensure their staying separate,

  Number of ways in which this event can be accomplished

  	=	Number of ways in which nDP places can be filled with nDL letters
  Or	=	Number of permutations or arrangements with nDP items taking nDL at a time

  ⇒ nSE2

  =

  nDPPnDL

The number of words that can be formed

Examples

The number of words that can be formed with the letters of the word DIALOGUE such that no two consonants come together