What is that probability can the letters of the word story be arranged so that T and Y are always together will be?

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Find the number of ways letters of the word HISTORY can be arranged if Y and T are together.

There are 7 letters in the word HISTORYWhen ‘Y’ and ‘T’ are together.Let us consider ‘Y’ and ‘T’ as one unitThis unit with the other 5 letters is to be arranged.

∴ The number of arrangements of one unit and 5 letters = 6P6 = 6!

Also, ‘Y’ and ‘T’ can be arranged among themselves in 2P2 i.e., 2! ways.∴ Total number of arrangements when Y and T are always together = 6! × 2! = 720 × 2 = 1440

∴ 1440 words can be formed if Y and T are together.

Concept: Permutations - Permutations When Repetitions Are Allowed

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Answer

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Hint:In this problem we will arrange all the letters by taking T and Y as a single unit and after that we will find out the total possible arrangements of these two letters T and Y.

Complete step-by-step answer:

Given: we have given the word ‘STORY’ and out of this word we have to find out that in how many ways can the word ‘STORY’ be arranged while the letter T and V come together.First, find the letters available in the given word which are 5.Now, we will take the letter T and Y as a single unit.So, the total number of letters present in the given word after taking the letter T and Y as a single unit.

Now, that means there are 4 letters in the word.So, total arrangements present in it is given by ${\text{N}}!$Here ${\text{N}}$is the total units which are 4.Then, substitute the value of ${\text{N}}$ in ${\text{N}}!$$  {\text{N}}! = 4! \\    \Rightarrow 4 \times 3 \times 2 \times 1 \\    \Rightarrow 24 \\ $And now, we can arrange letters T and Y by $2!$ ways.So, to get the total arrangements , we will multiply both the situations.So, total arrangements is ,Arrangements of 4 letters $ \times $arrangements of two letter T and Y.That is ,$4! \times 2!$Now, solve the equation.$ = 24 \times 2 = 48$So, the total ways of arrangements are 48 ways.Hence, the option D is the correct answer.

Note:Make sure that you consider the alphabets T and Y as one single unit. Considering them as two separate units will lead to wrong answers. Also remember to see that the position of the alphabets is proper in such problems.