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Let Tn denote the number of triangles which can be formed using the vertices of a regular polygon of n sides. If Tn + 1 - Tn = 21, then n equals (2001S)
- a)5
- b)7
- c)6
- d)4
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
Let Tn denote the number of triangles which can be formed using the ve...
Solution:
Given, Tn - 1 - Tn = 21
Let's try to find a general formula for Tn.
Consider a regular polygon of n sides. To form a triangle, we need to choose 3 vertices out of n vertices. The number of ways to choose 3 vertices out of n vertices is given by the combination formula.
Number of ways to choose 3 vertices out of n vertices = C(n, 3) = n!/[(n-3)!*3!]
Number of triangles that can be formed using the vertices of a regular polygon of n sides = Tn = C(n, 3)
Therefore, Tn = n!/[(n-3)!*3!]
Now, let's use this formula to solve the given equation.
Tn - 1 - Tn = 21
n!/[(n-4)!*3!] - n!/[(n-3)!*3!] = 21
n(n-3)/6 - (n-3)/2 = 21
Simplifying the above equation, we get
n^2 - 9n + 66 = 0
Solving the above quadratic equation, we get n = 6 or 3.
But n cannot be 3 as a polygon cannot have less than 3 sides.
Therefore, n = 6.
Hence, the correct option is (B) 7.
Given, Tn - 1 - Tn = 21
Let's try to find a general formula for Tn.
Consider a regular polygon of n sides. To form a triangle, we need to choose 3 vertices out of n vertices. The number of ways to choose 3 vertices out of n vertices is given by the combination formula.
Number of ways to choose 3 vertices out of n vertices = C(n, 3) = n!/[(n-3)!*3!]
Number of triangles that can be formed using the vertices of a regular polygon of n sides = Tn = C(n, 3)
Therefore, Tn = n!/[(n-3)!*3!]
Now, let's use this formula to solve the given equation.
Tn - 1 - Tn = 21
n!/[(n-4)!*3!] - n!/[(n-3)!*3!] = 21
n(n-3)/6 - (n-3)/2 = 21
Simplifying the above equation, we get
n^2 - 9n + 66 = 0
Solving the above quadratic equation, we get n = 6 or 3.
But n cannot be 3 as a polygon cannot have less than 3 sides.
Therefore, n = 6.
Hence, the correct option is (B) 7.
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Community Answer
Let Tn denote the number of triangles which can be formed using the ve...
As per question,
⇒ n(n - 1) (n + 1 -n + 2)= 126
⇒ n (n – 1) = 42 ⇒ n (n – 1) = 7 × 6 ⇒ n = 7.
⇒ n (n – 1) = 42 ⇒ n (n – 1) = 7 × 6 ⇒ n = 7.
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Let Tn denote the number of triangles which can be formed using the vertices of a regular polygon of n sides. If Tn + 1 - Tn = 21, then n equals (2001S)a)5b)7c)6d)4Correct answer is option 'B'. Can you explain this answer?
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