Answer
Verified
425.7k+ views
Hint: The number of ways of choosing $r$ things from $n$ things is given by ${}^{n}{{C}_{r}}$.
The value of ${}^{n}{{C}_{r}}$ is given as ${}^{n}{{C}_{r}}=\dfrac{n!}{\left( n-r \right)!r!}$.
Given- Total number of children $=11$.
Total number of mangoes $=5$.
Total number of bananas $=2$.
Total number of apples $=4$.
We need to find the number of ways to distribute the fruits among the children such that each child gets one fruit.
Now , instead of choosing fruits to divide among the children , we will choose children to whom each fruit is distributed . This is done because each child is different and there will be $11$ ways to choose each child independently , but each fruit is similar in a group , i.e. each mango is similar to another , each apple is similar to another . So , there will be one way to choose one mango from \[5\] mangoes.
So , we need to choose $5$ children for mangoes, $4$ children for apples and $2$ children for bananas.
We know, number of ways of choosing $r$ things from $n$ things is given by ${}^{n}{{C}_{r}}$.
And we also know ${}^{n}{{C}_{r}}=\dfrac{n!}{\left( n-r \right)!r!}$
So , number of ways of choosing $5$ children for mangoes from $11$ children $={}^{11}{{C}_{5}}$
So , remaining children $=11-5=6$
Now, $4$ children are to be chosen from $6$ children for apples. So , number of ways $={}^{6}{{C}_{4}}$
Remaining number of children $=6-4=2$
Number of ways of choosing $2$ children from $2$ children for bananas $={}^{2}{{C}_{2}}$
Now, all the selections are dependent on each other. So , total number of ways of choosing
\[\begin{align}
& ={}^{11}{{C}_{5}}\times {}^{6}{{C}_{4}}\times {}^{2}{{C}_{2}} \\
& =\dfrac{11!}{5!\left( 11-5 \right)!}\times \dfrac{6!}{4!\left( 6-4 \right)!}\times \dfrac{2!}{\left( 2-2 \right)!2!} \\
& =\dfrac{11!}{5!{6!}}\times \dfrac{{6!}}{{2!}4!}\times \dfrac{{2!}}{0!2!} \\
& =\dfrac{11!}{5!}\times \dfrac{1}{4!}\times \dfrac{1}{0!2!} \\
\end{align}\]
Now, we know $\dfrac{a!}{b!}=\left( a \right)\times \left( a-1 \right)\times \left( a-2 \right)\times .......\times \left( a-b \right)$
And $0!=1$
So,
$\begin{align}
& =\dfrac{11!}{5!}\times \dfrac{1}{4!}\times \dfrac{1}{0!2!}=11\times 10\times 9\times 8\times 7\times 6\times \dfrac{1}{4\times 3\times 2\times 1}\times \dfrac{1}{1\times 2\times 1} \\
& =11\times 10\times 9\times 7 \\
& =77\times 90 \\
& =6930 \\
\end{align}$
Hence , the number of ways of distributing $5$ mangoes, $4$ apples and $2$ bananas among $11$ children is equal to \[6930\].
Note: Alternative Solution:
We know , the number of ways of distributing \[n\] things such that \[p\] are of one kind , \[q\] are of second kind and \[r\] are of third kind is given by \[\dfrac{n!}{p!\times q!\times r!}\]. So , the number of ways of distributing $5$ mangoes, $4$ apples and $2$ bananas among $11$ children is given by \[\dfrac{11!}{5!\times 4!\times 2!}=6930\] .
Hence , the number of ways of distributing $5$ mangoes, $4$ apples and $2$ bananas among $11$ children is equal to \[6930\].
The value of ${}^{n}{{C}_{r}}$ is given as ${}^{n}{{C}_{r}}=\dfrac{n!}{\left( n-r \right)!r!}$.
Given- Total number of children $=11$.
Total number of mangoes $=5$.
Total number of bananas $=2$.
Total number of apples $=4$.
We need to find the number of ways to distribute the fruits among the children such that each child gets one fruit.
Now , instead of choosing fruits to divide among the children , we will choose children to whom each fruit is distributed . This is done because each child is different and there will be $11$ ways to choose each child independently , but each fruit is similar in a group , i.e. each mango is similar to another , each apple is similar to another . So , there will be one way to choose one mango from \[5\] mangoes.
So , we need to choose $5$ children for mangoes, $4$ children for apples and $2$ children for bananas.
We know, number of ways of choosing $r$ things from $n$ things is given by ${}^{n}{{C}_{r}}$.
And we also know ${}^{n}{{C}_{r}}=\dfrac{n!}{\left( n-r \right)!r!}$
So , number of ways of choosing $5$ children for mangoes from $11$ children $={}^{11}{{C}_{5}}$
So , remaining children $=11-5=6$
Now, $4$ children are to be chosen from $6$ children for apples. So , number of ways $={}^{6}{{C}_{4}}$
Remaining number of children $=6-4=2$
Number of ways of choosing $2$ children from $2$ children for bananas $={}^{2}{{C}_{2}}$
Now, all the selections are dependent on each other. So , total number of ways of choosing
\[\begin{align}
& ={}^{11}{{C}_{5}}\times {}^{6}{{C}_{4}}\times {}^{2}{{C}_{2}} \\
& =\dfrac{11!}{5!\left( 11-5 \right)!}\times \dfrac{6!}{4!\left( 6-4 \right)!}\times \dfrac{2!}{\left( 2-2 \right)!2!} \\
& =\dfrac{11!}{5!{6!}}\times \dfrac{{6!}}{{2!}4!}\times \dfrac{{2!}}{0!2!} \\
& =\dfrac{11!}{5!}\times \dfrac{1}{4!}\times \dfrac{1}{0!2!} \\
\end{align}\]
Now, we know $\dfrac{a!}{b!}=\left( a \right)\times \left( a-1 \right)\times \left( a-2 \right)\times .......\times \left( a-b \right)$
And $0!=1$
So,
$\begin{align}
& =\dfrac{11!}{5!}\times \dfrac{1}{4!}\times \dfrac{1}{0!2!}=11\times 10\times 9\times 8\times 7\times 6\times \dfrac{1}{4\times 3\times 2\times 1}\times \dfrac{1}{1\times 2\times 1} \\
& =11\times 10\times 9\times 7 \\
& =77\times 90 \\
& =6930 \\
\end{align}$
Hence , the number of ways of distributing $5$ mangoes, $4$ apples and $2$ bananas among $11$ children is equal to \[6930\].
Note: Alternative Solution:
We know , the number of ways of distributing \[n\] things such that \[p\] are of one kind , \[q\] are of second kind and \[r\] are of third kind is given by \[\dfrac{n!}{p!\times q!\times r!}\]. So , the number of ways of distributing $5$ mangoes, $4$ apples and $2$ bananas among $11$ children is given by \[\dfrac{11!}{5!\times 4!\times 2!}=6930\] .
Hence , the number of ways of distributing $5$ mangoes, $4$ apples and $2$ bananas among $11$ children is equal to \[6930\].
Recently Updated Pages
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE
Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE
What are the possible quantum number for the last outermost class 11 chemistry CBSE
Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Write an application to the principal requesting five class 10 english CBSE
Difference Between Plant Cell and Animal Cell
a Tabulate the differences in the characteristics of class 12 chemistry CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
Discuss what these phrases mean to you A a yellow wood class 9 english CBSE
List some examples of Rabi and Kharif crops class 8 biology CBSE