Answer
Verified
392.1k+ views
Hint: Here first we will find the number of subsets of the integers 1, 2, 3,……..m containing three elements which contain the integer 5. Then we will form an equation according to the given data and solve for the value of m to get the required answer.
Complete step-by-step answer:
It is given that:-
One quarter of all the subsets containing three elements of the integers 1, 2, 3,……..m contain the integer 5.
Now the subsets of the integers containing three elements are given by:- \[^m{C_3}\]
Also, number of subsets of integers containing three elements in which one element is 5 given by:- \[^{m - 1}{C_2}\]
Now it is given that, 1/4th of the all the subsets containing three elements of the integers 1, 2, 3,……..m contain the integer 5.
This implies,
\[\dfrac{1}{4}\left( {^m{C_3}} \right){ = ^{m - 1}}{C_2}\]
Now we will use the following formula:-
\[^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\]
Applying this formula we get:-
\[\dfrac{1}{4}\left( {\dfrac{{m!}}{{3!\left( {m - 3} \right)!}}} \right) = \dfrac{{\left( {m - 1} \right)!}}{{2!\left( {m - 1 - 2} \right)!}}\]
Solving it further we get:-
\[\dfrac{1}{4}\left( {\dfrac{{m \times \left( {m - 1} \right)!}}{{3 \times 2!\left( {m - 3} \right)!}}} \right) = \dfrac{{\left( {m - 1} \right)!}}{{2!\left( {m - 3} \right)!}}\]
Cancelling the terms we get:-
\[\dfrac{1}{4}\left( {\dfrac{m}{3}} \right) = 1\]
Now solving for m we get:-
\[
\dfrac{m}{{12}} = 1 \\
\Rightarrow m = 12 \\
\]
Therefore, the value of m is 12.
Hence, option (1) is the correct option.
Note: Students can also solve the given problem by finding the difference between the number of subsets of integers containing three elements and the number of three element subsets of integers which do not contain element 5.
Number of subsets of integers containing three elements in which one element is 5 = number of subsets of integers containing three elements - the number of three element subsets of integers which do not contain element 5
Hence, we get:
\[{\text{Number of subsets of integers containing three elements in which one element is 5}}{ = ^m}{C_3}{ - ^{m - 1}}{C_3}\]
Complete step-by-step answer:
It is given that:-
One quarter of all the subsets containing three elements of the integers 1, 2, 3,……..m contain the integer 5.
Now the subsets of the integers containing three elements are given by:- \[^m{C_3}\]
Also, number of subsets of integers containing three elements in which one element is 5 given by:- \[^{m - 1}{C_2}\]
Now it is given that, 1/4th of the all the subsets containing three elements of the integers 1, 2, 3,……..m contain the integer 5.
This implies,
\[\dfrac{1}{4}\left( {^m{C_3}} \right){ = ^{m - 1}}{C_2}\]
Now we will use the following formula:-
\[^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\]
Applying this formula we get:-
\[\dfrac{1}{4}\left( {\dfrac{{m!}}{{3!\left( {m - 3} \right)!}}} \right) = \dfrac{{\left( {m - 1} \right)!}}{{2!\left( {m - 1 - 2} \right)!}}\]
Solving it further we get:-
\[\dfrac{1}{4}\left( {\dfrac{{m \times \left( {m - 1} \right)!}}{{3 \times 2!\left( {m - 3} \right)!}}} \right) = \dfrac{{\left( {m - 1} \right)!}}{{2!\left( {m - 3} \right)!}}\]
Cancelling the terms we get:-
\[\dfrac{1}{4}\left( {\dfrac{m}{3}} \right) = 1\]
Now solving for m we get:-
\[
\dfrac{m}{{12}} = 1 \\
\Rightarrow m = 12 \\
\]
Therefore, the value of m is 12.
Hence, option (1) is the correct option.
Note: Students can also solve the given problem by finding the difference between the number of subsets of integers containing three elements and the number of three element subsets of integers which do not contain element 5.
Number of subsets of integers containing three elements in which one element is 5 = number of subsets of integers containing three elements - the number of three element subsets of integers which do not contain element 5
Hence, we get:
\[{\text{Number of subsets of integers containing three elements in which one element is 5}}{ = ^m}{C_3}{ - ^{m - 1}}{C_3}\]
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
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Summary of the poem Where the Mind is Without Fear class 8 english CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Write an application to the principal requesting five class 10 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
What is the z value for a 90 95 and 99 percent confidence class 11 maths CBSE