How many ways can the letters of the word mobile be arranged so as to begin with vowel and end with consonants *?

Nội dung chính

Nội dung chính

• In how many ways can be the letter of the word ‘STRANGE’ be arranged so that(a) The vowel may appear in the odd places(b) The vowels are never separated(c) The vowels never come together
• How many ways can the letter of the word MATHEMATICS be arranged so that the vowels always come together?
• How many ways can the letters of the word strange be arranged so that the vowels never come together?
• How many arrangements are there of the letters from brains in which the vowels are together?
• How many ways can the letters of the word machine be arranged so that the vowels may occupy only the odd positions *?
• How many ways can the letters of the word machine be arranged so that the vowels may occupy only the odd positions * 210 576 144 456?
• In what ways the letters of the word machine can be arranged so that the vowels occupy only the even positions?
• How many different ways can the letters of the word computer be arranged so that vowels always come together?
• How many ways Word arrange can be arranged in which vowels are together?

• In how many ways can be the letter of the word ‘STRANGE’ be arranged so that(a) The vowel may appear in the odd places(b) The vowels are never separated(c) The vowels never come together
• How many ways can the letter of the word MATHEMATICS be arranged so that the vowels always come together?
• How many ways can the letters of the word strange be arranged so that the vowels never come together?
• How many arrangements are there of the letters from brains in which the vowels are together?
• How many ways can the letters of the word machine be arranged so that the vowels may occupy only the odd positions *?

In how many ways can be the letter of the word ‘STRANGE’ be arranged so that(a) The vowel may appear in the odd places(b) The vowels are never separated(c) The vowels never come together

Answer

Hint:There are 7 letters in the word ‘STRANGE’ in which two are vowels. There are four odd places. So, we have to find the total number of ways for these four places. The other 5 letters may be placed in the five places in 5! ways. By this we get the total number of required arrangements. If vowels are never separated, in this case consider all the vowels a single letter. If all the vowels never come together in this case, we subtract all vowels together from the total number of arrangements.Complete step-by-step answer:
For finding number of ways of placing n things in r objects, we use ${}^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}$. Also, for arranging n objects without restrictions, we have $n!$ ways.
(a) There are 7 letters in the word ‘STRANGE’ amongst which 2 are vowels and there are 4 odd places (1, 3, 5, 7) where these two vowels are to be placed together.
Total number of ways can be expressed by replacing n = 4 and r = 2,
$\begin{align}
  & {}^{4}{{P}_{2}}=\dfrac{4!}{\left( 4-2 \right)!} \\
 & =\dfrac{4!}{2!} \\
 & =\dfrac{4\times 3\times 2!}{2!} \\
 & =4\times 3=12 \\
\end{align}$
And corresponding to these 12 ways the other 5 letters may be placed in 5! ways $=5\times 4\times 3\times 2\times 1=120$.
Therefore, the total number of required arrangements = $12\times 120=1440$ ways.

(b) Now, vowels are not to be separated. So, we consider all vowels as a single letter.
There are six letters S, T, R, N, G, (AE) , so they can arrange themselves in 6! ways and two vowels can arrange themselves in 2! ways.
Total number of required arrangements \[=6!\times 2!=6\times 5\times 4\times 3\times 2\times 1\times 2\times 1=1440\] ways.

(c) Now, for the number of arrangements when all vowels never come together, we subtract the total arrangement where all vowels have occurred together from the total number of arrangements.
The total number of arrangements = 7! = 5040 ways
And the number of arrangements in which the vowels do not come together $=7!-6!2!$
number of arrangements in which the vowels do not come together $=5040 -1440 = 3600$ ways.
Note: Knowledge of permutation and arrangement of objects under some restriction and without restriction is must for solving this problem. Students must be careful while making the possible cases. They must consider all the permutations in word like in part (b), a common mistake is done by not considering the arrangements of vowels.

hello students today occasion is in how many ways can the letter of the word combine Kabir arranged so that vowels are never separated for we have the word combine a m e i n so we have total 7 letter 1 2 3 4 5 6 7 letters a year and we have to find the number of ways in which ones are never separated so there are a total 7 letters therefore number of possible permutations is equal to 7 factorial so we can write year total number of possible permutation per Mein stations is equal to 7 factorial and there are three were we have three were here so we

Android oi koormai and 4 consonant so we can write here for consonant better c b and so we have consider all the vowels Subah aap to consider all the vowels a single set letter and there then we are left with 5 letters in total we will permit permit 5 letters and later replaced the single viable y3 actual mobile written adjacent to one another so the number of ways to arrange these five letter is so we can write a number of ways to arrange these five letter is

is 5 factorial also 3 evil that that will come together can be arranged in and we can write a mobil3 actual was written here is also three Evil that come together that will come to Gadar 2 for this we get 3 factorial hair bacteria based total number of permutation so we have to find the total number of number of permutation

of given letter of given letter combine such that such that all mobile with come together all bubble will come together to this is a quest to find factorial in 23 factorial so this is equals to 722 this is our answer thank you

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In how many ways can the letters of the word MANIFOLD be arranged so that the vowels are separated?

I tried total permutations in which vowels are together, which gives 36000 which was wrong.

asked Aug 28, 2017 at 7:24

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All letters are distinct, so there are $8!$ permutations. This count both good and bad ones.

Consider two of the vowels as one letter ($3!/1!$ cases). Each of the cases constitutes $7!$ bad permutations.

Consider three vowels as on letter ($3!$ cases). Each of the cases constitute $6!$ bad permutations which were however counted twice by the $3! 7!$ above, so these bust be added.

I.e. $8!-3!7!+3!6!$

answered Aug 28, 2017 at 7:35

CoolwaterCoolwater

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

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

How many ways can the letters of the word strange be arranged so that the vowels never come together?

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

How many arrangements are there of the letters from brains in which the vowels are together?

Hence, 302400 different arrangements can be made when all vowels are together. 2) All the vowels are not together. First arrange the consonants then arrange the vowels. After arranging 9 consonants there 10 places will remain blank.

How many ways can the letters of the word machine be arranged so that the vowels may occupy only the odd positions *?

In how many different ways can the letters of the word 'MACHINE' be arranged so that the vowels may occupy only the odd positions ? Now, 3 vowels can be placed at any of the three places, out of the four marked 1, 3, 5,7. Number of ways of arranging the vowels = 4P3 = (4 *3 * 2) = 24.

How many ways can the letters of the word machine be arranged so that the vowels may occupy only the odd positions * 210 576 144 456?

Required number of ways = (360 * 2) = 720. In how many different ways can the letters of the word 'MACHINE' be arranged so that the vowels may occupy only the odd positions ?

In what ways the letters of the word machine can be arranged so that the vowels occupy only the even positions?

Total ways of arranging vowels = 6*4=24. There are 4 blank places & 4 consonants so they can change their positions in 4! = 24 ways. Total number of ways of arranging letters of a given word = 24^2=576.

How many different ways can the letters of the word computer be arranged so that vowels always come together?

Similarly, there is only 1 way to arrange the vowels in the remaining 3 positions. Therefore, there are 56 ways to arrange COMPUTER, given the above constraint. Alternatively we can choose to position the vowels first. We will get C(8,3) = 56 ways to select a combination of positions on which the vowels must stay.

How many ways Word arrange can be arranged in which vowels are 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 mobile be arranged so that the consonants always occupy the odd places?

How many ways the word mobile can be arranged so that vowels always come together?

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

How many words can be made out of the letters of the word mobile?