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?

This section covers permutations and combinations.

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• 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 words can be formed from MATHEMATICS if vowels are together?
• How many arrangements can be made from the letters of the word MATHEMATICS?
• How many permutations can be made from the letters of the word MATHEMATICS if the vowels are to be together?
• How many vowels are there in the word MATHEMATICS?

Arranging Objects

The number of ways of arranging n unlike objects in a line is n! (pronounced ‘n factorial’). n! = n × (n – 1) × (n – 2) ×…× 3 × 2 × 1

Example

How many different ways can the letters P, Q, R, S be arranged?

The answer is 4! = 24.

This is because there are four spaces to be filled: _, _, _, _

The first space can be filled by any one of the four letters. The second space can be filled by any of the remaining 3 letters. The third space can be filled by any of the 2 remaining letters and the final space must be filled by the one remaining letter. The total number of possible arrangements is therefore 4 × 3 × 2 × 1 = 4!

• The number of ways of arranging n objects, of which p of one type are alike, q of a second type are alike, r of a third type are alike, etc is:

n!        .
p! q! r! …

Example

In how many ways can the letters in the word: STATISTICS be arranged?

There are 3 S’s, 2 I’s and 3 T’s in this word, therefore, the number of ways of arranging the letters are:

10!=50 400
3! 2! 3!

Rings and Roundabouts

• The number of ways of arranging n unlike objects in a ring when clockwise and anticlockwise arrangements are different is (n – 1)!

When clockwise and anti-clockwise arrangements are the same, the number of ways is ½ (n – 1)!

Example

Ten people go to a party. How many different ways can they be seated?

Anti-clockwise and clockwise arrangements are the same. Therefore, the total number of ways is ½ (10-1)! = 181 440

Combinations

The number of ways of selecting r objects from n unlike objects is:

Example

There are 10 balls in a bag numbered from 1 to 10. Three balls are selected at random. How many different ways are there of selecting the three balls?

10C3 =10!=10 × 9 × 8= 120
             3! (10 – 3)!3 × 2 × 1

Permutations

A permutation is an ordered arrangement.

• The number of ordered arrangements of r objects taken from n unlike objects is:

nPr =       n!       .
          (n – r)!

Example

In the Match of the Day’s goal of the month competition, you had to pick the top 3 goals out of 10. Since the order is important, it is the permutation formula which we use.

10P3 =10!
            7!

= 720

There are therefore 720 different ways of picking the top three goals.

Probability

The above facts can be used to help solve problems in probability.

Example

In the National Lottery, 6 numbers are chosen from 49. You win if the 6 balls you pick match the six balls selected by the machine. What is the probability of winning the National Lottery?

The number of ways of choosing 6 numbers from 49 is 49C6 = 13 983 816 .

Therefore the probability of winning the lottery is 1/13983816 = 0.000 000 071 5 (3sf), which is about a 1 in 14 million chance.

Mathematic can be arranged in 453,600 different ways if it is ten letters and only use each letter once. Assuming all vowels will be together 15,120 arrangements.

Requires work with permutations and factorials. Factorial is written as '!' . Factorial is the multiplication of all it lower terms.
Eg 10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1

Mathematic has ten letters and as such can be arranged 10! ways. As it has repeating letters you divide by this repetitions.

As such it becomes #(10!)/(2!*2!*2!)# . This equation equals 453 600

For the second part it should be treated as two parts. All the vowels are grouped together so there are effectively only 7 letters left.

This you would write as #(7!)/(2!*2!)# (Lose a 2! as the repeating vowel is not counted.) Then for the vowel, it is #(4!)/(2!)#.
You now multiply these together to get your answer of 15120

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 \[2{\text{ }} \ldots \ldots .,{\text{ }}{n_k}\]identical objects of type $k$ is \[\dfrac{{n!}}{{{n_1}!\,{n_2}!.......{n_k}!}}\]

Complete step by step solution:
Word Mathematics has $11$ letters
\[\mathop {\text{M}}\limits^{\text{1}} \mathop {\text{A}}\limits^{\text{2}} \mathop {\text{T}}\limits^{\text{3}} \mathop {\text{H}}\limits^{\text{4}} \mathop {\text{E}}\limits^{\text{5}} \mathop {\text{M}}\limits^{\text{6}} \mathop {\text{A}}\limits^{\text{7}} \mathop {\text{T}}\limits^{\text{8}} \mathop {\text{I}}\limits^{\text{9}} \mathop {{\text{ C}}}\limits^{{\text{10}}} \mathop {{\text{ S}}}\limits^{{\text{11}}} \]
In which M, A, T are repeated twice.
By using the formula \[\dfrac{{n!}}{{{n_1}!\,{n_2}!.......{n_k}!}}\], first, we have to find the number of ways in which the word ‘Mathematics’ can be written is
$
  P = \dfrac{{11!}}{{2!2!2!}} \\
   = \dfrac{{11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{2 \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} = \dfrac{{8!}}{{2!2!}} \\
   = \dfrac{{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{2 \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
\[
  {P_2} = \dfrac{{4!}}{{2!}} \\
   = \dfrac{{4 \times 3 \times 2 \times 1}}{{2 \times 1}} \\
   = 12 \\
 \]
$\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 words can be formed from MATHEMATICS if vowels are together?

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

How many arrangements can be made from the letters of the word MATHEMATICS?

Complete step-by-step answer: 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 permutations can be made from the letters of the word MATHEMATICS if the vowels are to be together?

Mathematic can be arranged in 453,600 different ways if it is ten letters and only use each letter once. Assuming all vowels will be together 15,120 arrangements.

How many vowels are there in the word MATHEMATICS?

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. Now, vowels A, E, I, A, has 4 letters in which A occurs 2 times and rest are different.

How many arrangements can be made with the letters of the word MATHEMATICS if all vowels don't occur together?

How many ways can the letters of the word MATHEMATICS be arranged so that the vowels?

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

How many ways can the letters of the word MATHEMATICS be arranged so that the vowels come together Brainly?