JEE Mains · Maths · STD 11 - 6. permutation and combination
Consider a rectangle \(ABCD\) having \(5,7,6,9\) points in the interior of the line segments \(AB,CD , BC , DA\) respectively. Let \(\alpha\) be the number of triangles having these points from different sides as vertices and \(\beta\) be the number of quadrilaterals having these points from different sides as vertices. Then \((\beta-\alpha)\) is equal to :
- A \(795\)
- B \(1173\)
- C \(1890\)
- D \(717\)
Answer & Solution
Correct Answer
(D) \(717\)
Step-by-step Solution
Detailed explanation
\(\alpha=\) Number of triangles \(\alpha=5 \cdot 6 \cdot 7+5 \cdot 7 \cdot 9+5 \cdot 6 \cdot 9+6 \cdot 7 \cdot 9\) \(\quad=210+315+270+378\) \(\quad=1173\) \(\beta=\) Number of Quadrilateral \(\beta=5 \cdot 6 \cdot 7 \cdot 9=1890\) \(\beta-\alpha=1890-1173=717\)
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- Let \(\mathrm{f}(\mathrm{x})=\cos \left(2 \tan ^{-1} \sin \left(\cot ^{-1} \sqrt{\frac{1-\mathrm{x}}{\mathrm{x}}}\right)\right)\) \(0<\mathrm{x}<1\). Then :JEE Mains 2021 Hard
- If \(f(x)=\int \frac{1}{x^{1 / 4}\left(1+x^{1 / 4}\right)} \mathrm{d} x, f(0)=-6\), then \(f(1)\) is equal to :JEE Mains 2025 Medium
- The coefficient of \(x^2\) in the expansion of \(\left(2x^2 + \dfrac{1}{x}\right)^{10}\), \(x \neq 0\), is :JEE Mains 2026 Easy
- If \(y = \tan^{-1}\left(\dfrac{3\cos x - 4\sin x}{4\cos x + 3\sin x}\right) + 2\tan^{-1}\left(\dfrac{x}{1+\sqrt{1-x^2}}\right)\), then \(\dfrac{dy}{dx}\) at \(x = \dfrac{\sqrt{3}}{2}\) is equal to:JEE Mains 2026 Medium
- Let the domains of the functions
\(\mathrm{f}(\mathrm{x})=\log _4 \log _3 \log _7\left(8-\log _2\left(\mathrm{x}^2+4 \mathrm{x}+5\right)\right)\) and \(g(x)=\sin ^{-1}\left(\frac{7 x+10}{x-2}\right)\) be \((\alpha, \beta)\) and \([\gamma, \delta]\), respectively. Then \(\alpha^2+\beta^2+\gamma^2+\delta^2\) is equal to :-JEE Mains 2025 Medium - \(\int {\frac{{{{\sin }^8}\,x - {{\cos }^8}\,x}}{{\left( {1 - 2\,{{\sin }^2}\,x\,{{\cos }^2}\,x} \right)}}} dx\) is equal toJEE Mains 2014 Hard
More PYQs from JEE Mains
- If \(\sum_{r=1}^{50} \tan ^{-1} \frac{1}{2 r^{2}}=p\), then the value of \(\tan p\) is :JEE Mains 2021 Hard
- If \(\vec{a}=2 \hat{i}+\hat{j}+2 \hat{k},\) then the value of \(|\hat{ i } \times(\overrightarrow{ a } \times \hat{ i })|^{2}+|\hat{j} \times(\overrightarrow{ a } \times \hat{ j })|^{2}+|\hat{ k } \times(\overrightarrow{ a } \times \hat{ k })|^{2}\) is equal toJEE Mains 2020 Medium
- The system of linear equation \(x + y + z = 2, 2x + 3y + 2z = 5\), \(2x + 3y + (a^2 -1)\,z = a + 1\) thenJEE Mains 2019 Hard
- If \(\cot \alpha=1\) and \(\sec \beta=-\frac{5}{3}\), where \(\pi<\alpha<\frac{3 \pi}{2}\) and \(\frac{\pi}{2}<\beta<\pi\), then the value of \(\tan (\alpha+\beta)\) and the quadrant in which \(\alpha+\beta\) lies, respectively areJEE Mains 2022 Medium
- Points \(P (-3,2), Q (9,10)\) and \(R (\alpha, 4)\) lie on a circle \(C\) with \(P R\) as its diameter. The tangents to \(C\) at the points \(Q\) and \(R\) intersect at the point \(S\). If \(S\) lies on the line \(2 x - ky =1\), then \(k\) is equal to \(.........\).JEE Mains 2023 Hard
- If \(\overrightarrow{ a }=\hat{ i }+2 \hat{ k }, \overrightarrow{ b }=\hat{ i }+\hat{ j }+\hat{ k }, \overrightarrow{ c }=7 \hat{ i }-3 \hat{ k }+4 \hat{ k }\) \(\overrightarrow{ r } \times \overrightarrow{ b }+\overrightarrow{ b } \times \overrightarrow{ c }=\overrightarrow{0}\) and \(\overrightarrow{ r } \cdot \overrightarrow{ a }=0\) then \(\overrightarrow{ r } \cdot \overrightarrow{ c }\) is equal to :JEE Mains 2023 Hard