930 Deepawali greeting cards are exchanged amongst the students of a class. If every student sends a card to every other student then what is the number of students in the class?
Answer
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Hint: In this question apply the concept of combination i.e. when one student sends a card to another student and that student sends a card to the previous one then the number of possible ways is 2, so use this concept to reach the solution of the question.
Let us consider there are no students in the class.
So, if every students sends a card to every other student i.e. A gives a card to B and B gives a card to A, so between two students there are two ways of distributing the cards, so, among n students according to combination property the number of possible ways is $2{}^n{C_2}$.
(Two multiplied by ${}^n{C_2}$ is because cards are exchanged amongst all students i.e. every student gives cards to every other student if not then two is not multiplied).
And it is given that there are 930 cards exchanged amongst the students of a class.
$ \Rightarrow 2{}^n{C_2} = 930$…………….. (1)
Now we all know that $^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$, so use this property in above equation we have,
${}^n{C_2} = \dfrac{{n!}}{{2!\left( {n - 2} \right)!}}$
Now we all know that $n! = n\left( {n - 1} \right)\left( {n - 2} \right)!$, so use this property in above equation we have,
${}^n{C_2} = \dfrac{{n\left( {n - 1} \right)\left( {n - 2} \right)!}}{{2!\left( {n - 2} \right)!}} = \dfrac{{n\left( {n - 1} \right)}}{{2 \times 1}}$
Therefore from equation (1) we have,
$ \Rightarrow 2\dfrac{{n\left( {n - 1} \right)}}{{2 \times 1}} = 930$
$
\Rightarrow {n^2} - n = 930 \\
\Rightarrow {n^2} - n - 930 = 0 \\
$
Now factorize this equation we have,
$
\Rightarrow {n^2} - 31n + 30n - 930 = 0 \\
\Rightarrow n\left( {n - 31} \right) + 30\left( {n - 31} \right) = 0 \\
\Rightarrow \left( {n + 30} \right)\left( {n - 31} \right) = 0 \\
\Rightarrow \left( {n + 30} \right) = 0,{\text{ }}\left( {n - 31} \right) = 0 \\
\Rightarrow n = - 30,{\text{ 31}} \\
$
But n should not be negative.
So the number of students in the class is 31.
So, this is the required answer.
Note: In such types of questions the key concept we have to remember is that when the cards is exchanged among n students then number of possible ways is $2{}^n{C_2}$, then apply the property of combination and construct the equation as above and simplify, we will get the required number of students in the class.
Let us consider there are no students in the class.
So, if every students sends a card to every other student i.e. A gives a card to B and B gives a card to A, so between two students there are two ways of distributing the cards, so, among n students according to combination property the number of possible ways is $2{}^n{C_2}$.
(Two multiplied by ${}^n{C_2}$ is because cards are exchanged amongst all students i.e. every student gives cards to every other student if not then two is not multiplied).
And it is given that there are 930 cards exchanged amongst the students of a class.
$ \Rightarrow 2{}^n{C_2} = 930$…………….. (1)
Now we all know that $^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$, so use this property in above equation we have,
${}^n{C_2} = \dfrac{{n!}}{{2!\left( {n - 2} \right)!}}$
Now we all know that $n! = n\left( {n - 1} \right)\left( {n - 2} \right)!$, so use this property in above equation we have,
${}^n{C_2} = \dfrac{{n\left( {n - 1} \right)\left( {n - 2} \right)!}}{{2!\left( {n - 2} \right)!}} = \dfrac{{n\left( {n - 1} \right)}}{{2 \times 1}}$
Therefore from equation (1) we have,
$ \Rightarrow 2\dfrac{{n\left( {n - 1} \right)}}{{2 \times 1}} = 930$
$
\Rightarrow {n^2} - n = 930 \\
\Rightarrow {n^2} - n - 930 = 0 \\
$
Now factorize this equation we have,
$
\Rightarrow {n^2} - 31n + 30n - 930 = 0 \\
\Rightarrow n\left( {n - 31} \right) + 30\left( {n - 31} \right) = 0 \\
\Rightarrow \left( {n + 30} \right)\left( {n - 31} \right) = 0 \\
\Rightarrow \left( {n + 30} \right) = 0,{\text{ }}\left( {n - 31} \right) = 0 \\
\Rightarrow n = - 30,{\text{ 31}} \\
$
But n should not be negative.
So the number of students in the class is 31.
So, this is the required answer.
Note: In such types of questions the key concept we have to remember is that when the cards is exchanged among n students then number of possible ways is $2{}^n{C_2}$, then apply the property of combination and construct the equation as above and simplify, we will get the required number of students in the class.
Last updated date: 21st Sep 2023
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