Question

how many words can formed from the letter of words " DAUGHTER " SO that vowels never come together ​

Answer 1

$Heya\:$

➡️There are 3 vowels and 5 consonants.

➡️I first arranged 5 consonants in five places in 5! ways.

⏩6 gaps are created.

✅....6C3 Out of these 6 gaps, I selected 3 gaps in ways and then made the vowels permute in those 3 selected places in 3! ways.

✔️This leads me to my answer

$5!⋅6C3⋅3!=14400$

Answer 2

$\huge\bf\pink{\mid{\overline{\underline{Your\: Answer}}}\mid}$

_____________________

__________

$\underline{\underline{\huge\mathfrak{14,400\:words}}}$

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___________

step-by-step explanation :

Total number of consonants = 5

and,

Total number of vowels = 3

Now,

Now vowel should come together,

so,

we have to fill them in the gaps

which can be done in this way :

(__)C1(__)C2(__)C3(__)C4(__)C5(__)

here,

C1.....C5 denotes the consonants

and

(__) represent the place where vowels will come.

Total no. of (__) = 6

Now,

The no. of ways to arrange the consonants = 5!

and,

the no. of ways to arrange vowels = p(6,3)

Therefore,

total number of words possible are

= 5! × p(6,3)

= (6! × 5!)/(6-3)!

= 6! × 5 × 4 × 3!/ 3!

= 6! × 5 × 4

= 14,400

Hence,

14,400 words are possible.