Answer
Verified
395.1k+ views
Hint: First of all, find the possible number of ways in which perfect square factors of \[{2^5},{3^6},{5^2}\] can be arranged individually. Then use the multiplicative principle of permutations to get the required answer. So, use this concept to reach the solution of the given problem.
Complete step-by-step answer:
For a perfect square, the power of each should be even.
The possible factors of \[{2^5}\] are \[{2^0},{2^1},{2^2},{2^3},{2^4},{2^5}\]
So, the possible perfect square factors of \[{2^5}\] are \[{2^0},{2^2},{2^4}\].
Therefore, possible number of ways of arranging the perfect square factors of \[{2^5}\] = 3
The possible factors of \[{3^6}\] are \[{3^0},{3^1},{3^2},{3^3},{3^4},{3^5},{3^6}\]
So, the possible perfect square factors of \[{3^6}\] are \[{3^0},{3^2},{3^4},{3^6}\].
Therefore, possible number of ways of arranging the perfect square factors of \[{3^6}\] = 4
The possible factors of \[{5^2}\] are \[{5^0},{5^1},{5^2}\]
So, the possible perfect square factors of \[{5^2}\] are \[{5^0},{5^2}\].
Therefore, possible number of ways of arranging the perfect square factors of \[{5^2}\] = 2
By using multiplicative principle of permutations, we have
The total number of ways of arranging the perfect square factors of \[{2^5} \times {3^6} \times {5^2}\] are \[3 \times 4 \times 2 = 24\]
Hence there are 24 factors of \[{2^5} \times {3^6} \times {5^2}\] which are perfect squares.
Thus, the correct option is A. 24
Note: In this problem we have used multiplicative principle permutations i.e., if there are \[x\] number of ways of arranging one thing and\[y\] number of ways of arranging another, then the total number of ways of arranging both the things is given in \[xy\] number of ways.
Complete step-by-step answer:
For a perfect square, the power of each should be even.
The possible factors of \[{2^5}\] are \[{2^0},{2^1},{2^2},{2^3},{2^4},{2^5}\]
So, the possible perfect square factors of \[{2^5}\] are \[{2^0},{2^2},{2^4}\].
Therefore, possible number of ways of arranging the perfect square factors of \[{2^5}\] = 3
The possible factors of \[{3^6}\] are \[{3^0},{3^1},{3^2},{3^3},{3^4},{3^5},{3^6}\]
So, the possible perfect square factors of \[{3^6}\] are \[{3^0},{3^2},{3^4},{3^6}\].
Therefore, possible number of ways of arranging the perfect square factors of \[{3^6}\] = 4
The possible factors of \[{5^2}\] are \[{5^0},{5^1},{5^2}\]
So, the possible perfect square factors of \[{5^2}\] are \[{5^0},{5^2}\].
Therefore, possible number of ways of arranging the perfect square factors of \[{5^2}\] = 2
By using multiplicative principle of permutations, we have
The total number of ways of arranging the perfect square factors of \[{2^5} \times {3^6} \times {5^2}\] are \[3 \times 4 \times 2 = 24\]
Hence there are 24 factors of \[{2^5} \times {3^6} \times {5^2}\] which are perfect squares.
Thus, the correct option is A. 24
Note: In this problem we have used multiplicative principle permutations i.e., if there are \[x\] number of ways of arranging one thing and\[y\] number of ways of arranging another, then the total number of ways of arranging both the things is given in \[xy\] number of ways.
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
Difference Between Plant Cell and Animal Cell
Select the word that is correctly spelled a Twelveth class 10 english CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
What is the z value for a 90 95 and 99 percent confidence class 11 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
What organs are located on the left side of your body class 11 biology CBSE
What is BLO What is the full form of BLO class 8 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE