Answer
Verified
423.9k+ views
Hint: First we need to find values of x, y and z such that their product is equal to 30. Then use permutation to calculate the number of possible arrangements of these numbers.
Complete step-by-step answer:
Given the problem, we need to find the number of positive integral solutions of $xyz = 30$.
It means that we need to find the permutation of the number of values of $x,y,z$such that the above equation is satisfied.
First, we need to find the factors of 30.
Factors of 30 are 1,2,3,5,6,10,15 and 30.
From the given factors, we need to find sets of 3 factors such that their product is 30.
The sets satisfying above conditions would be
$
x = 30,y = 1,z = 1 \\
x = 15,y = 2,z = 1 \\
x = 10,y = 3,z = 1 \\
x = 6,y = 5,z = 1 \\
x = 5,y = 3,z = 2 \\
$
The values of $x,y,z$can be interchanged as the multiplication is commutative in real numbers.
We know that number of ways to arrange n objects at a time $ = n!$
Also, the number of ways to arrange n objects at a time with r objects repeated $ = \dfrac{{n!}}{{r!}}$
Using the above permutation formula in the obtained sets, we get to find the number of viable or ways.
$
x = 30,y = 1,z = 1{\text{ , no}}{\text{. of ways = }}\dfrac{{3!}}{{2!}} = 3 \\
x = 15,y = 2,z = 1{\text{ , no}}{\text{. of ways = 3!}} = 6 \\
x = 10,y = 3,z = 1{\text{ , no}}{\text{. of ways = 3!}} = 6 \\
x = 6,y = 5,z = 1{\text{ , no}}{\text{. of ways = 3!}} = 6 \\
x = 5,y = 3,z = 2{\text{ , no}}{\text{. of ways = 3!}} = 6 \\
$
This gives a total no. of ways = 6+6+6+6+3=27.
Hence the total number of ways in which product of $xyz = 30$is 27.
Hence no. of positive integral solutions of $xyz = 30$is 27.
Therefore, option (B) 27 is the correct answer.
Note: While counting the possible combination of factors in the above problem, the factors could be chosen such that the factor on the left is always greater than or equal to the factor on its right, not less. This approach could help in saving time and rectifying the redundant cases. Formula used for arrangement of objects should be kept in mind while solving problems like above.
Complete step-by-step answer:
Given the problem, we need to find the number of positive integral solutions of $xyz = 30$.
It means that we need to find the permutation of the number of values of $x,y,z$such that the above equation is satisfied.
First, we need to find the factors of 30.
Factors of 30 are 1,2,3,5,6,10,15 and 30.
From the given factors, we need to find sets of 3 factors such that their product is 30.
The sets satisfying above conditions would be
$
x = 30,y = 1,z = 1 \\
x = 15,y = 2,z = 1 \\
x = 10,y = 3,z = 1 \\
x = 6,y = 5,z = 1 \\
x = 5,y = 3,z = 2 \\
$
The values of $x,y,z$can be interchanged as the multiplication is commutative in real numbers.
We know that number of ways to arrange n objects at a time $ = n!$
Also, the number of ways to arrange n objects at a time with r objects repeated $ = \dfrac{{n!}}{{r!}}$
Using the above permutation formula in the obtained sets, we get to find the number of viable or ways.
$
x = 30,y = 1,z = 1{\text{ , no}}{\text{. of ways = }}\dfrac{{3!}}{{2!}} = 3 \\
x = 15,y = 2,z = 1{\text{ , no}}{\text{. of ways = 3!}} = 6 \\
x = 10,y = 3,z = 1{\text{ , no}}{\text{. of ways = 3!}} = 6 \\
x = 6,y = 5,z = 1{\text{ , no}}{\text{. of ways = 3!}} = 6 \\
x = 5,y = 3,z = 2{\text{ , no}}{\text{. of ways = 3!}} = 6 \\
$
This gives a total no. of ways = 6+6+6+6+3=27.
Hence the total number of ways in which product of $xyz = 30$is 27.
Hence no. of positive integral solutions of $xyz = 30$is 27.
Therefore, option (B) 27 is the correct answer.
Note: While counting the possible combination of factors in the above problem, the factors could be chosen such that the factor on the left is always greater than or equal to the factor on its right, not less. This approach could help in saving time and rectifying the redundant cases. Formula used for arrangement of objects should be kept in mind while solving problems like above.
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