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Permutation is known as the process of organizing the group, body toàn thân, or numbers in order, selecting the body toàn thân or numbers from the set, is known as combinations in such a way that the order of the number does not matter.

Nội dung chính

  • In how many ways can the letters of the word IMPOSSIBLE be arranged so that all the vowels come together?
  • In how many words can the letters of word ‘Mathematics’ be arranged so that (i) vowels are together (ii) vowels are not together
  • How many ways can the letter MATHEMATICS be arranged so that the vowels always come together?
  • How many ways the word over expand can be arranged
    so that all vowels come together?
  • How many
    arrangements can be made by the letters of the word MATHEMATICS in how many of them vowels are i Together II not together?
  • How many words can be made from the word MATHEMATICS in which vowels are together?
  • How many ways word arrange can be arranged in which vowels are not together?
  • How many ways the word over expand can be arranged so that all vowels come together?
  • How many ways all vowels come together?
  • How many ways can the letters of the word Missouri be arranged so that all vowels do not occur?

Nội dung chính

  • In how many ways can the letters of the word IMPOSSIBLE be arranged so that all the vowels come together?
  • In how many words can the letters of word ‘Mathematics’ be arranged so that (i) vowels are together (ii) vowels are not together
  • How many ways can the letter MATHEMATICS be arranged so that the vowels always come together?
  • How many ways the word over expand can be arranged so that all vowels come together?
  • How many arrangements can be made by the letters of the word MATHEMATICS in how many of them vowels are i Together II not together?
  • How many words can be made from the word MATHEMATICS in which vowels are together?

In mathematics, permutation is also known as the process of organizing a group in which all the members of a group are arranged into some sequence or order. The process of
permuting is known as the repositioning of its components if the group is already arranged. Permutations take place, in almost every area of mathematics. They mostly appear when different commands on certain limited sets are considered.

Permutation Formula

In permutation r things are picked from a group of n things without any replacement. In this order of picking matter.

nPr = (n!)/(n – r)!

Here,

n = group size, the
total number of things in the group

r = subset size, the number of things to be selected from the group

Combination

A combination is a function of selecting the number from a set, such that (not like permutation) the order of choice doesn’t matter. In smaller cases, it is conceivable to count the number of combinations. The combination is known as the merging of n things taken k a time without repetition. In combination, the order doesn’t matter
you can select the items in any order. To those combinations in which re-occurrence is allowed, the terms k-selection or k-combination with replication are frequently used.

Combination Formula

In combination r things are picked from a set of n things and where the order of picking does not matter.

nCr = n!⁄((n-r)! r!)

Here,

n = Number of items in set

r = Number of things picked from the group

In how many ways can the letters of the word IMPOSSIBLE be arranged so that all the vowels come together?

Solution:

Vowels are: I,I,O,E

If all the vowels must come together then treat all the vowels as one super letter, next note the letter ‘S’ repeats so we’d use

7!/2! = 2520 

Now count the ways the vowels in the super
letter can be arranged, since there are 4 and 1 2-letter(I’i) repeat the super letter of vowels would be arranged in 12 ways i.e., (4!/2!)

= (7!/2! × 4!/2!) 

= 2520(12)

= 30240 ways

Similar Questions

Question 1: In how many ways can the letters be arranged so that all the vowels came together word is CORPORATION?

Solution:

Vowels are :- O,O,A,I,O

If all the vowels must
come together then treat all the vowels as one super letter, next note the R’r letter repeat so we’d use

7!/2! = 2520

Now count the ways the vowels in the super letter can be arranged, since there are 5 and 1 3-letter repeat the super letter of vowels would be arranged in 20 ways i.e., (5!/3!)

= (7!/2! × 5!/3!)

= 2520(20)

= 50400 ways

Question 2: In how many different ways can the letters of the word ‘MATHEMATICS’ be arranged such that the
vowels must always come together?

Solution:

Vowels are :- A,A,E,I

Next, treat the block of vowels like a single letter, let’s just say V for vowel. So then we have MTHMTCSV – 8 letters, but 2 M’s and 2 T’s. So there are

8!/2!2! = 10,080

Now count the ways the vowels letter can be arranged, since there are 4 and 1 2-letter repeat the super letter of vowels would be arranged in 12 ways i.e., (4!/2!)

= (8!/2!2! × 4!/2!)

= 10,080(12)

= 120,960 ways

Question 3: In How many ways the letters of the word RAINBOW be arranged in which vowels are never together?

Solution:

Vowels are :- A, I, O  

Consonants are:- R, N, B, W.

Arrange all the vowels in between the consonants so that they can not be together. There are 5 total places between the consonants. So, vowels can be organize in 5P3 ways and the four consonants
can be organize in 4! ways.

Therefore, the total arrangements are 5P3 * 4! = 60 * 24 = 1440

In how many words can the letters of word ‘Mathematics’ be arranged so that (i) vowels are together (ii) vowels are not together

Answer

Verified

Hint: First find the number of ways in which word ‘Mathematics’ can be written,
and then we use permutation formula with repetition which is given as under,
Number of permutation of $n$objects with$n$, identical objects of type$1,n_2$identical objects of type [2text ldots ldots .,text n_k]identical objects of type $k$ is [dfracn!n_1!,n_2!…….n_k!]

Complete step by step solution:
Word Mathematics has $11$ letters
[mathop textMlimits^text1 mathop textAlimits^text2 mathop textTlimits^text3
mathop textHlimits^text4 mathop textElimits^text5 mathop textMlimits^text6 mathop textAlimits^text7 mathop textTlimits^text8 mathop textIlimits^text9 mathop text Climits^text10 mathop text Slimits^text11 ]
In which M, A, T are repeated twice.
By using the formula [dfracn!n_1!,n_2!…….n_k!], first, we have to find the number of ways in which the word ‘Mathematics’
can be written is
$
  P = dfrac11!2!2!2! \
   = dfrac11 times 10 times 9 times 8 times 7 times 6 times 5 times 4 times 3 times 2 times 12 times 1 times 2 times 1 times 2 times 1 \
   = 11 times 10 times 9 times 7 times 6 times 5 times 4 times 3 times 2 times 1 \
   = 4989600 \
 $
In [4989600]distinct ways, the letter of the word ‘Mathematics’ can be written.

(i) When vowels are taken together:
In the word ‘Mathematics’, we treat the vowels A, E, A, I as one letter. Thus, we have MTHMTCS (AEAI).
Now, we have to arrange letters, out of which M occurs twice, T occurs twice, and the rest are different.
$therefore $Number of ways of arranging the word ‘Mathematics’ when consonants are occurring together
$
  P_1 = dfrac8!2!2! \
   = dfrac8 times 7 times 6 times 5 times 4 times 3 times 2
times 12 times 1 times 2 times 1 \
   = 10080 \
 $
Now, vowels A, E, I, A, has $4$ letters in which A occurs $2$ times and rest are different.
$therefore $Number of arranging the letter
[
  P_2 = dfrac4!2! \
   = dfrac4 times 3 times 2 times 12 times 1 \
   = 12 \
 ]
$therefore $Per a number of words $ = (10080) times (12)$
In which vowel come
together $ = 120960$ways

(ii) When vowels are not taken together:
When vowels are not taken together then the number of ways of arranging the letters of the word ‘Mathematics’ are
$
   = 4989600 – 120960 \
   = 4868640 \
 $

Note: In this type of question, we use the permutation formula for a word in which the letters are repeated. Otherwise, simply solve the question by counting the number of letters of the word it has and
in case of the counting of vowels, we will consider the vowels as a single unit.

How many ways can the letter MATHEMATICS be arranged so that the vowels always come together?

∴ Required number of words = (10080 x 12) = 120960.

How many ways the word over expand can be arranged
so that all vowels come together?

The word EXTRA can be arranged in such a way that the vowels will be together = 4! × 2! The letters of the words EXTRA be arranged so that the vowels are never together = (120 – 48) = 72 ways. ∴ The letters of the words EXTRA be arranged so that the vowels are never together in 72 ways.

How many
arrangements can be made by the letters of the word MATHEMATICS in how many of them vowels are i Together II not together?

The word MATHEMATICS consists of 2 M’s, 2 A’s, 2 T’s, 1 H, 1 E, 1 I, 1 C and 1 S. Therefore, a total of 4989600 words can be formed using all the letters of the word MATHEMATICS.

How many words can be made from the word MATHEMATICS in which vowels are together?

Total no of cases in which the word MATHEMATICS can be written = 11! = 8! Hence, the number of words can be made by using all letters of the word MATHEMATICS in which all vowels are never together is 378000.

How many ways word arrange can be arranged in which vowels are not together?

number of arrangements in which the vowels do not come together =5040−1440=3600 ways.

How many ways the word over expand can be arranged so that all vowels come together?

The word EXTRA can be arranged in such a way that the vowels will be together = 4! × 2! The letters of the words EXTRA be arranged so that the vowels are never together = (120 – 48) = 72 ways. ∴ The letters of the words EXTRA be arranged so that the vowels are never together in 72 ways.

How many ways all vowels come together?

The number of ways the word TRAINER can be arranged so that the vowels always come together are 360.

How many ways can the letters of the word Missouri be arranged so that all vowels do not occur?

In the word MISSOURI, the letters S and I are repeated twice and hence we can not use this in the arrangement of letters. If we ignore the repetition of the letters, the total distinct letters to be arranged are 6, that is, M, I, S, O, U and R. Hence, the number of permutations possible = 6! =720.
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