Answer
415.5k+ views
Hint: First we will check the total number of ways to select the 12 letters of the English alphabets by the formula n! Then, we will fix the positions of A and B and then we will calculate the probability of 4 letters lying between A and B by the formula ${}^n{C_r}$where n is the total ways and r is the selected ways in which we are required to calculate the probability. Then we will calculate the probability of the required arrangement by the formula: possible ways / total number of ways.
Complete step-by-step answer:
First of all, we are given 12 letters of English alphabet and they are chosen randomly. Given that A and B are two letters from them, we need to find the probability that there are 4 letters between A and B.
Let us first calculate the total number of ways to select 12 random alphabets. The total number of ways will be n! i. e., 12!
Now, we are told that A and B are two letters from the 12 selected. So, there are 10 remaining alphabets now. Let us say A and B are given two positions such that there are 4 letters in between them. So, the number of ways to do so will be 2!
The 4 letters which are between A and B can be selected from the remaining 10 by the formula ${}^n{C_r}$ so, the total number of ways to select 4 letters will be ${}^{10}{C_4} \times 4!$
Now, we can see that these events occur simultaneously after each other. So, the total number of ways will be the product of all the above events i.e., the ways will be $2! \times 4! \times {}^{10}{C_4}$.
There will be remaining 6 letters now and let us assume that the arrangement we just calculated as one individual letter. So, we can say that there are 7 letters now in total. So, the total number of ways to arrange them will be $7! \times 2! \times 4! \times {}^{10}{C_4}$.
Therefore, the required probability will be equal to $\dfrac{{7! \times 2! \times 4! \times {}^{10}{C_4}}}{{12!}}$by the formula of required probability = total possible ways / total number of ways.
Hence, we get $probability = \dfrac{{7!2!4!10!}}{{12!6!4!}} = \dfrac{7}{{66}}$ .
Hence, the probability of A and B having 4 letters in between is 7/66.
Therefore, option(D) is correct.
Note: You should take care of using formulae and especially while deducing the number of ways for 4 letters to lie between A and B. You may get confused in the final steps where you need to use multiplication by thinking that addition should be used. These events have occurred one after and they complete the probability, so they are bound to be multiplied instead of adding.
Complete step-by-step answer:
First of all, we are given 12 letters of English alphabet and they are chosen randomly. Given that A and B are two letters from them, we need to find the probability that there are 4 letters between A and B.
Let us first calculate the total number of ways to select 12 random alphabets. The total number of ways will be n! i. e., 12!
Now, we are told that A and B are two letters from the 12 selected. So, there are 10 remaining alphabets now. Let us say A and B are given two positions such that there are 4 letters in between them. So, the number of ways to do so will be 2!
The 4 letters which are between A and B can be selected from the remaining 10 by the formula ${}^n{C_r}$ so, the total number of ways to select 4 letters will be ${}^{10}{C_4} \times 4!$
Now, we can see that these events occur simultaneously after each other. So, the total number of ways will be the product of all the above events i.e., the ways will be $2! \times 4! \times {}^{10}{C_4}$.
There will be remaining 6 letters now and let us assume that the arrangement we just calculated as one individual letter. So, we can say that there are 7 letters now in total. So, the total number of ways to arrange them will be $7! \times 2! \times 4! \times {}^{10}{C_4}$.
Therefore, the required probability will be equal to $\dfrac{{7! \times 2! \times 4! \times {}^{10}{C_4}}}{{12!}}$by the formula of required probability = total possible ways / total number of ways.
Hence, we get $probability = \dfrac{{7!2!4!10!}}{{12!6!4!}} = \dfrac{7}{{66}}$ .
Hence, the probability of A and B having 4 letters in between is 7/66.
Therefore, option(D) is correct.
Note: You should take care of using formulae and especially while deducing the number of ways for 4 letters to lie between A and B. You may get confused in the final steps where you need to use multiplication by thinking that addition should be used. These events have occurred one after and they complete the probability, so they are bound to be multiplied instead of adding.
Recently Updated Pages
How do you integrate intcos 3left x right class 10 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
How do you integrate dfracleft lnx right2x2 class 10 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
How do you integrate dfrac3x35x211x+9x22x3 using partial class 10 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Integrate 2xdx class 10 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
How do you integrate 10xdx from 1 to 0 class 10 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Integers is not closed under A Addition B Subtraction class 10 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Which of the following is the most stable ecosystem class 12 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which of the following is the most stable ecosystem class 12 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write an application to the principal requesting five class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
The term vaccine was introduced by A Jenner B Koch class 12 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)