Question

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. 1890
B. 795
C. 717
D. 1173

Answer 1

Hint: Here in this question, we have to find the number of triangles and quadrilaterals that can be formed by the points 5, 6, 7 and 9. To find the number of ways we use permutation and combination concepts. Since this question involves selection of points, we use the combination concept.

Complete step by step solution: 
Selections and combinations are synonyms. Combinations represent the choosing of items from a predetermined group of items. We're not trying to arrange anything here. We're going to pick them. We use the symbol \[{}^{n}{{C}_{r}}\]to represent the number of distinct r-selections or combinations among a set of n objects.
Firstly we calculate \[\alpha \]
\[\alpha \]be the number of triangles with these vertices coming from various sides.
We have four points, but in order to create triangles, we must select three of them. Thus, we have
\[\alpha ={}^{6}{{C}_{1}}{}^{7}{{C}_{1}}{}^{9}{{C}_{1}}+{}^{5}{{C}_{1}}{}^{7}{{C}_{1}}{}^{9}{{C}_{1}}+{}^{5}{{C}_{1}}{}^{6}{{C}_{1}}{}^{9}{{C}_{1}}+{}^{5}{{C}_{1}}{}^{6}{{C}_{1}}{}^{7}{{C}_{1}}\]
On applying the formula we have
\[\Rightarrow \alpha =6.7.9+5.7.9+5.6.9+5.6.7\]
\[\Rightarrow \alpha =378+315+270+210\]
\[\Rightarrow \alpha =1173\]
Now we will find the value of \[\beta \]
\[\beta \] be the number of quadrilaterals with these points from various sides as vertices will now be calculated.
From the available 4, we must select 4 points.
\[\beta ={}^{5}{{C}_{1}}{}^{6}{{C}_{1}}{}^{7}{{C}_{1}}{}^{9}{{C}_{1}}\]
\[\Rightarrow \beta =5.6.7.9\]
\[\Rightarrow \beta =1890\]
Now \[(\beta -\alpha )=1890-1173\]
On subtracting we get
\[(\beta -\alpha )=717\]
Therefore the value of \[(\beta -\alpha )\] is 717.
So the correct answer is option D

Note: The student should know the formula for the permutation and combination. The formula for the permutation is \[{}^{n}{{P}_{r}}=\dfrac{n!}{r!}\] and the formula for the combination is \[{}^{n}{{C}_{r}}=\dfrac{n!}{(n-r)!r!}\]. When solving permutation problems, we know about the factorial; it means that the function multiplies a number by every number below it.