Answer
Verified
403.8k+ views
Hint: We will solve this question with the help of combinations and hence we will use the formula \[{}^{n}{{C}_{r}}=\dfrac{n!}{r!(n-r)!}\]. Non collinear means points are not in a line and are random whereas concyclic means if we draw the circle using three points it should pass from the 4th point.
Complete step-by-step answer:
Now it is mentioned in the question that 3 points are not collinear and 4 points are concyclic and the total number of points given is 10.
So we can get circles from joining 3 non collinear points out of 10 but this also includes circles formed from joining 3 points out of 4 concyclic points and hence we will subtract these circles because they are repetitive. And now we will add 1 as there is a circle which would pass all 4 points. Hence using this information, we get,
Total number of circles \[={}^{10}{{C}_{3}}-{}^{4}{{C}_{3}}+1.......(1)\]
Now applying the formula \[{}^{n}{{C}_{r}}=\dfrac{n!}{r!(n-r)!}\] in equation (1), we get,
\[\Rightarrow \dfrac{10!}{3!(10-3)!}-\dfrac{4!}{3!(4-3)!}+1.......(2)\]
Now expanding the factorials and solving equation (2) we get,
\[\Rightarrow \dfrac{10\times 9\times 8\times 7!}{3\times 2\times 7!}-\dfrac{4\times 3!}{3!}+1.......(3)\]
Now cancelling similar terms in equation (3), we get,
\[\begin{align}
& \Rightarrow \dfrac{10\times 9\times 8}{3\times 2}-4+1 \\
& \Rightarrow 10\times 3\times 4-4+1=120-4+1=117 \\
\end{align}\]
Hence the number of different circles is 117. So the correct answer is option (c).
Note: A combination is a way to order or arrange a set or number of things uniquely. And knowing this formula \[{}^{n}{{C}_{r}}=\dfrac{n!}{r!(n-r)!}\] is important . We can make a mistake in solving equation (1) if we do not properly expand \[{}^{10}{{C}_{3}}\] and \[{}^{4}{{C}_{3}}\]. Also understanding the concept of collinear and concyclic is important.
Complete step-by-step answer:
Now it is mentioned in the question that 3 points are not collinear and 4 points are concyclic and the total number of points given is 10.
So we can get circles from joining 3 non collinear points out of 10 but this also includes circles formed from joining 3 points out of 4 concyclic points and hence we will subtract these circles because they are repetitive. And now we will add 1 as there is a circle which would pass all 4 points. Hence using this information, we get,
Total number of circles \[={}^{10}{{C}_{3}}-{}^{4}{{C}_{3}}+1.......(1)\]
Now applying the formula \[{}^{n}{{C}_{r}}=\dfrac{n!}{r!(n-r)!}\] in equation (1), we get,
\[\Rightarrow \dfrac{10!}{3!(10-3)!}-\dfrac{4!}{3!(4-3)!}+1.......(2)\]
Now expanding the factorials and solving equation (2) we get,
\[\Rightarrow \dfrac{10\times 9\times 8\times 7!}{3\times 2\times 7!}-\dfrac{4\times 3!}{3!}+1.......(3)\]
Now cancelling similar terms in equation (3), we get,
\[\begin{align}
& \Rightarrow \dfrac{10\times 9\times 8}{3\times 2}-4+1 \\
& \Rightarrow 10\times 3\times 4-4+1=120-4+1=117 \\
\end{align}\]
Hence the number of different circles is 117. So the correct answer is option (c).
Note: A combination is a way to order or arrange a set or number of things uniquely. And knowing this formula \[{}^{n}{{C}_{r}}=\dfrac{n!}{r!(n-r)!}\] is important . We can make a mistake in solving equation (1) if we do not properly expand \[{}^{10}{{C}_{3}}\] and \[{}^{4}{{C}_{3}}\]. Also understanding the concept of collinear and concyclic is important.
Recently Updated Pages
Basicity of sulphurous acid and sulphuric acid are
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
What is the stopping potential when the metal with class 12 physics JEE_Main
The momentum of a photon is 2 times 10 16gm cmsec Its class 12 physics JEE_Main
Using the following information to help you answer class 12 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