Answer
64.8k+ views
Hint: First we find out the number of ways the distribution can be done in all the 3 cases. The cases being 6 distinct colored balls or 6 balls three each of two colors or 6 balls of two each of three colors. We find the number of ways each case can be done and add them to find the final answer.
Complete step-by-step solution
A box contains 6 balls which may be all of different colors or three each of two colors or two each of three different colors.
So, the number of cases of possible options is 3.
Case 1: 6 distinct colored balls
We need to choose 3 balls out of 6 distinct ones which will be done in ${}^{6}{{C}_{3}}=\dfrac{6!}{3!\times 3!}=20$.
Case 2: 6 balls three each of two colors
We need to choose 3 balls. The choices can be 2 same-colored balls and 1 different or all three of same-colored balls.
The first part (2 white 1 black or 2 black 1 white) can be done in
$2\left[ {}^{3}{{C}_{2}}\times {}^{3}{{C}_{1}} \right]=2\times \dfrac{3!}{2!\times 1!}\times \dfrac{3!}{2!\times 1!}=18$.
The second part (3 white or 3 black) can be done in ${}^{3}{{C}_{3}}+{}^{3}{{C}_{3}}=1+1=2$.
Case 3: 6 balls of two each of three colors
We need to choose 3 balls. The choices can be 2 same-colored balls and 1 different or all three of different colored balls.
The first part can be done in choosing which color we are taking 2 balls off.
\[{}^{3}{{C}_{1}}\times {}^{2}{{C}_{1}}\times {}^{2}{{C}_{1}}=3\times 2\times 2=12\].
The second part (1 white and 1 black and 1 red) can be done in \[{}^{2}{{C}_{1}}\times {}^{2}{{C}_{1}}\times {}^{2}{{C}_{1}}={{2}^{3}}=8\].
Total number of options are $20+18+2+12+8=60$. The value of N is 20.
Note: We need to separately find the ways of choosing and we can use a table and actual color names to solve the problem. The multiplication of choosing is happening as the events are independent of each other.
Complete step-by-step solution
A box contains 6 balls which may be all of different colors or three each of two colors or two each of three different colors.
So, the number of cases of possible options is 3.
Case 1: 6 distinct colored balls
We need to choose 3 balls out of 6 distinct ones which will be done in ${}^{6}{{C}_{3}}=\dfrac{6!}{3!\times 3!}=20$.
Case 2: 6 balls three each of two colors
We need to choose 3 balls. The choices can be 2 same-colored balls and 1 different or all three of same-colored balls.
The first part (2 white 1 black or 2 black 1 white) can be done in
$2\left[ {}^{3}{{C}_{2}}\times {}^{3}{{C}_{1}} \right]=2\times \dfrac{3!}{2!\times 1!}\times \dfrac{3!}{2!\times 1!}=18$.
The second part (3 white or 3 black) can be done in ${}^{3}{{C}_{3}}+{}^{3}{{C}_{3}}=1+1=2$.
Case 3: 6 balls of two each of three colors
We need to choose 3 balls. The choices can be 2 same-colored balls and 1 different or all three of different colored balls.
The first part can be done in choosing which color we are taking 2 balls off.
\[{}^{3}{{C}_{1}}\times {}^{2}{{C}_{1}}\times {}^{2}{{C}_{1}}=3\times 2\times 2=12\].
The second part (1 white and 1 black and 1 red) can be done in \[{}^{2}{{C}_{1}}\times {}^{2}{{C}_{1}}\times {}^{2}{{C}_{1}}={{2}^{3}}=8\].
Total number of options are $20+18+2+12+8=60$. The value of N is 20.
Note: We need to separately find the ways of choosing and we can use a table and actual color names to solve the problem. The multiplication of choosing is happening as the events are independent of each other.
Recently Updated Pages
Write a composition in approximately 450 500 words class 10 english JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Arrange the sentences P Q R between S1 and S5 such class 10 english JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
What is the common property of the oxides CONO and class 10 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
What happens when dilute hydrochloric acid is added class 10 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
If four points A63B 35C4 2 and Dx3x are given in such class 10 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
The area of square inscribed in a circle of diameter class 10 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Other Pages
Excluding stoppages the speed of a bus is 54 kmph and class 11 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
A boat takes 2 hours to go 8 km and come back to a class 11 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Electric field due to uniformly charged sphere class 12 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
According to classical free electron theory A There class 11 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
In the ground state an element has 13 electrons in class 11 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)