Answer
Verified
390.9k+ views
Hint: In this question it is given that the total number of handshakes is 28. So, if we consider that there are ‘n’ number of people. Then the first person shakes hands with (n-1) persons (he can’t shake hands with himself so we have subtracted 1 that n-1). Similarly, the second person shakes hands with (n-2) persons (n-2 because the second person can’t shake hands with himself and the first person is already counted), and so on. At last apply formula, total number of handshakes$\left( {n - 1} \right){\text{ }} + \left( {n - 2} \right){\text{ }} + \ldots .{\text{ }} + 1 = \dfrac{{n(n - 1)}}{2}$.
Complete step-by-step answer: Given,
Total handshakes were exchanged at the conclusion of a party = 28.
We have, total number of handshakes = $\left( {n - 1} \right){\text{ }} + \left( {n - 2} \right){\text{ }} + \ldots .{\text{ }} + 1 = \dfrac{{n(n - 1)}}{2}$
Using the above formula, we get;
$\dfrac{{n(n - 1)}}{2} = 28$
$ \Rightarrow n(n - 1) = 56$
\[ \Rightarrow {n^2} - n - 56 = 0\]
$ \Rightarrow {n^2} - 8n + 7n - 56 = 0$
$ \Rightarrow n(n - 8) + 7(n - 8) = 0$
$ \Rightarrow (n - 8)(n + 7) = 0$
$ \Rightarrow n = 8$ or $ \Rightarrow n = - 7$
Negative number of people is not possible. So, n = -7 is not acceptable.
Therefore, n = 8 is the only acceptable value.
So, the number of people present in the party = 8.
Note: In the above question first person shakes hand with 7 persons, second person shakes hand with 6 persons; similarly, third person shakes hand with 5 persons and so on.
Total handshakes = 7 + 6 + 5 + 4 + 3 + 2 + 1 = 28 Handshakes.
There is an alternative method which can be used in this question by using Combination$^n{C_2} = \dfrac{{n!}}{{2!(n - 2)!}} = \dfrac{{n(n - 1)}}{2}$.
Above equation has a numerical value of 28.
Thus, we get the equation;
$\dfrac{{n(n - 1)}}{2} = 28$
Solving this equation, we get two values, choose the appropriate value.
Subsequent steps are already covered in the main answer. This is the alternative method. We can follow any of the steps for solving this question.
Complete step-by-step answer: Given,
Total handshakes were exchanged at the conclusion of a party = 28.
We have, total number of handshakes = $\left( {n - 1} \right){\text{ }} + \left( {n - 2} \right){\text{ }} + \ldots .{\text{ }} + 1 = \dfrac{{n(n - 1)}}{2}$
Using the above formula, we get;
$\dfrac{{n(n - 1)}}{2} = 28$
$ \Rightarrow n(n - 1) = 56$
\[ \Rightarrow {n^2} - n - 56 = 0\]
$ \Rightarrow {n^2} - 8n + 7n - 56 = 0$
$ \Rightarrow n(n - 8) + 7(n - 8) = 0$
$ \Rightarrow (n - 8)(n + 7) = 0$
$ \Rightarrow n = 8$ or $ \Rightarrow n = - 7$
Negative number of people is not possible. So, n = -7 is not acceptable.
Therefore, n = 8 is the only acceptable value.
So, the number of people present in the party = 8.
Note: In the above question first person shakes hand with 7 persons, second person shakes hand with 6 persons; similarly, third person shakes hand with 5 persons and so on.
Total handshakes = 7 + 6 + 5 + 4 + 3 + 2 + 1 = 28 Handshakes.
There is an alternative method which can be used in this question by using Combination$^n{C_2} = \dfrac{{n!}}{{2!(n - 2)!}} = \dfrac{{n(n - 1)}}{2}$.
Above equation has a numerical value of 28.
Thus, we get the equation;
$\dfrac{{n(n - 1)}}{2} = 28$
Solving this equation, we get two values, choose the appropriate value.
Subsequent steps are already covered in the main answer. This is the alternative method. We can follow any of the steps for solving this question.
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 Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE
Select the correct plural noun from the given singular class 10 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
The sum of three consecutive multiples of 11 is 363 class 7 maths CBSE
What is the z value for a 90 95 and 99 percent confidence class 11 maths CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
How many squares are there in a chess board A 1296 class 11 maths CBSE