Question:
The number of triangles whose vertices are at the vertices of an octagon but none of whose side happen to come from the sides of the octagon is
The number of triangles whose vertices are at the vertices of an octagon but none of whose side happen to come from the sides of the octagon is
Updated On: Jul 7, 2022
- 24
- 52
- 48
- 16
Hide Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Number of all possible triangles
= Number of selections of 3 points from 8 vertices
$=^8C_3=56$
Number of triangle with one side common with octagon $= 8 ? 4 = 32$
(Consider side $A_1A_2$. Since two points $A_3, A_8$ are adjacent, 3rd point should be chosen from remaining 4 points.)
Number of triangles having two sides common with octagon : All such triangles have three consecutive vertices, viz., $A_1A_2A_3, A_2A_3A_4, ..... A_8A_1A_2.$
Number of such triangles $= 8$
$\therefore$ Number of triangles with no side common $= 56 - 32 - 8 = 16.$
Was this answer helpful?
0
0
Top Questions on Permutations
- In how many ways can 5 people be arranged in a row?
- MHT CET - 2025
- Mathematics
- Permutations
- In a group of 3 girls and 4 boys, there are two boys \( B_1 \) and \( B_2 \). The number of ways in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but \( B_1 \) and \( B_2 \) are not adjacent to each other, is:
- JEE Main - 2025
- Mathematics
- Permutations
Among the 5 married couples, if the names of 5 men are matched with the names of their wives randomly, then the probability that no man is matched with the name of his own wife is ?
- TS EAMCET - 2024
- Mathematics
- Permutations
- If all the numbers which are greater than 6000 and less than 10000 are formed with the digits \( 3,5,6,7,8 \) without repetition of the digits, then the difference between the number of odd numbers and the number of even numbers among them is:
- TS EAMCET - 2024
- Mathematics
- Permutations
- The rank of the word "TABLE" counted from the rank of the word "BLATE" in dictionary order is:
- AP EAMCET - 2024
- Mathematics
- Permutations
View More Questions
Notes on Permutations
Concepts Used:
Permutations
A permutation is an arrangement of multiple objects in a particular order taken a few or all at a time. The formula for permutation is as follows:
\(^nP_r = \frac{n!}{(n-r)!}\)
nPr = permutation
n = total number of objects
r = number of objects selected
Types of Permutation
- Permutation of n different things where repeating is not allowed
- Permutation of n different things where repeating is allowed
- Permutation of similar kinds or duplicate objects