119290 If \(\frac{2}{9 !}+\frac{2}{3 ! 7 !}+\frac{1}{5 ! 5 !}=\frac{2^a}{b !}\) where \(a, b \in N\), then the ordered pair \((a, b)\) is

119293 If the sides \(\mathrm{AB}, \mathrm{BC}\) and \(\mathrm{CA}\) of a triangle \(\triangle \mathrm{ABC}\) have 3,5 and 6 interior points respectively, then the total number of triangles that can be constructed using these points as vertices, is equal to

119294 Consider a rectangle \(\mathrm{ABCD}\) having 5, 7, 6, 9 points in the interior of the line segments \(A B\), 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, \((\boldsymbol{\beta}-\boldsymbol{\alpha})\) is equal to

119295 Let \(n>2\) be an integer. Suppose that there are \(n\) metro stations in a city located along a circular path. Each pair of stations is connected a straight track only. Further, each pair of nearest stations is connected by blue line, whereas all remaining pairs of stations are connected by red line. If the number of red lines in 99 times the number of blue lines, then the value of \(n\) is

119296 The set \(S=\{1,2,3, \ldots . .12\}\) is to be partitoned into three sets \(A, B\) and \(C\) of equal size.
Thus, \(A \cup B \cup C=S\)
\(A \cap B=B \cap C=A \cap C=\phi\)
The number of ways to partition \(S\) is

119290 If \(\frac{2}{9 !}+\frac{2}{3 ! 7 !}+\frac{1}{5 ! 5 !}=\frac{2^a}{b !}\) where \(a, b \in N\), then the ordered pair \((a, b)\) is

119293 If the sides \(\mathrm{AB}, \mathrm{BC}\) and \(\mathrm{CA}\) of a triangle \(\triangle \mathrm{ABC}\) have 3,5 and 6 interior points respectively, then the total number of triangles that can be constructed using these points as vertices, is equal to

119294 Consider a rectangle \(\mathrm{ABCD}\) having 5, 7, 6, 9 points in the interior of the line segments \(A B\), 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, \((\boldsymbol{\beta}-\boldsymbol{\alpha})\) is equal to

119295 Let \(n>2\) be an integer. Suppose that there are \(n\) metro stations in a city located along a circular path. Each pair of stations is connected a straight track only. Further, each pair of nearest stations is connected by blue line, whereas all remaining pairs of stations are connected by red line. If the number of red lines in 99 times the number of blue lines, then the value of \(n\) is

119296 The set \(S=\{1,2,3, \ldots . .12\}\) is to be partitoned into three sets \(A, B\) and \(C\) of equal size.
Thus, \(A \cup B \cup C=S\)
\(A \cap B=B \cap C=A \cap C=\phi\)
The number of ways to partition \(S\) is

119290 If \(\frac{2}{9 !}+\frac{2}{3 ! 7 !}+\frac{1}{5 ! 5 !}=\frac{2^a}{b !}\) where \(a, b \in N\), then the ordered pair \((a, b)\) is

119293 If the sides \(\mathrm{AB}, \mathrm{BC}\) and \(\mathrm{CA}\) of a triangle \(\triangle \mathrm{ABC}\) have 3,5 and 6 interior points respectively, then the total number of triangles that can be constructed using these points as vertices, is equal to

119294 Consider a rectangle \(\mathrm{ABCD}\) having 5, 7, 6, 9 points in the interior of the line segments \(A B\), 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, \((\boldsymbol{\beta}-\boldsymbol{\alpha})\) is equal to

119295 Let \(n>2\) be an integer. Suppose that there are \(n\) metro stations in a city located along a circular path. Each pair of stations is connected a straight track only. Further, each pair of nearest stations is connected by blue line, whereas all remaining pairs of stations are connected by red line. If the number of red lines in 99 times the number of blue lines, then the value of \(n\) is

119296 The set \(S=\{1,2,3, \ldots . .12\}\) is to be partitoned into three sets \(A, B\) and \(C\) of equal size.
Thus, \(A \cup B \cup C=S\)
\(A \cap B=B \cap C=A \cap C=\phi\)
The number of ways to partition \(S\) is

119290 If \(\frac{2}{9 !}+\frac{2}{3 ! 7 !}+\frac{1}{5 ! 5 !}=\frac{2^a}{b !}\) where \(a, b \in N\), then the ordered pair \((a, b)\) is

119293 If the sides \(\mathrm{AB}, \mathrm{BC}\) and \(\mathrm{CA}\) of a triangle \(\triangle \mathrm{ABC}\) have 3,5 and 6 interior points respectively, then the total number of triangles that can be constructed using these points as vertices, is equal to

119294 Consider a rectangle \(\mathrm{ABCD}\) having 5, 7, 6, 9 points in the interior of the line segments \(A B\), 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, \((\boldsymbol{\beta}-\boldsymbol{\alpha})\) is equal to

119295 Let \(n>2\) be an integer. Suppose that there are \(n\) metro stations in a city located along a circular path. Each pair of stations is connected a straight track only. Further, each pair of nearest stations is connected by blue line, whereas all remaining pairs of stations are connected by red line. If the number of red lines in 99 times the number of blue lines, then the value of \(n\) is

119296 The set \(S=\{1,2,3, \ldots . .12\}\) is to be partitoned into three sets \(A, B\) and \(C\) of equal size.
Thus, \(A \cup B \cup C=S\)
\(A \cap B=B \cap C=A \cap C=\phi\)
The number of ways to partition \(S\) is

119290 If \(\frac{2}{9 !}+\frac{2}{3 ! 7 !}+\frac{1}{5 ! 5 !}=\frac{2^a}{b !}\) where \(a, b \in N\), then the ordered pair \((a, b)\) is

119293 If the sides \(\mathrm{AB}, \mathrm{BC}\) and \(\mathrm{CA}\) of a triangle \(\triangle \mathrm{ABC}\) have 3,5 and 6 interior points respectively, then the total number of triangles that can be constructed using these points as vertices, is equal to

119294 Consider a rectangle \(\mathrm{ABCD}\) having 5, 7, 6, 9 points in the interior of the line segments \(A B\), 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, \((\boldsymbol{\beta}-\boldsymbol{\alpha})\) is equal to

119295 Let \(n>2\) be an integer. Suppose that there are \(n\) metro stations in a city located along a circular path. Each pair of stations is connected a straight track only. Further, each pair of nearest stations is connected by blue line, whereas all remaining pairs of stations are connected by red line. If the number of red lines in 99 times the number of blue lines, then the value of \(n\) is

119296 The set \(S=\{1,2,3, \ldots . .12\}\) is to be partitoned into three sets \(A, B\) and \(C\) of equal size.
Thus, \(A \cup B \cup C=S\)
\(A \cap B=B \cap C=A \cap C=\phi\)
The number of ways to partition \(S\) is