Advertisements
Advertisements
Question
A convex polygon has 44 diagonals. Find the number of its sides.
Advertisements
Solution
Let n be the number of sides in a polygon.
Since, Polygon of n sides has (nC2 – n) number of diagonals
∴ nC2 – n = 44
= `(n!)/(2!(n - 2)!)` – n = 44
= `(n(n - 1)(n - 2)!)/(2*(n - 2)!)` – n = 44
⇒ `(n(n - 1))/2` – n = 44
= `(n^2 - n - 2n)/2` = 44
⇒ n2 – 3n = 44
⇒ n2 – 3n – 88 = 0
= n2 – 11n + 8n – 88 = 0
⇒ n(n – 11) + 8(n – 11) = 0
= (n – 11(n + 8) = 0
∴ n = 11 and n = – 8 ....[∵ n ≠ – 8]
So n = 11
Hence, the required number of sides = 11.
APPEARS IN
RELATED QUESTIONS
Find the number of ways of selecting 9 balls from 6 red balls, 5 white balls and 5 blue balls if each selection consists of 3 balls of each colour.
Determine the number of 5 card combinations out of a deck of 52 cards if there is exactly one ace in each combination.
How many words, with or without meaning, each of 2 vowels and 3 consonants can be formed from the letters of the word DAUGHTER?
It is required to seat 5 men and 4 women in a row so that the women occupy the even places. How many such arrangements are possible?
A coin is tossed five times and outcomes are recorded. How many possible outcomes are there?
How many three-digit numbers are there?
How many odd numbers less than 1000 can be formed by using the digits 0, 3, 5, 7 when repetition of digits is not allowed?
How many 3-digit numbers are there, with distinct digits, with each digit odd?
How many four digit different numbers, greater than 5000 can be formed with the digits 1, 2, 5, 9, 0 when repetition of digits is not allowed?
If nC12 = nC5, find the value of n.
If 15Cr : 15Cr − 1 = 11 : 5, find r.
There are 10 professors and 20 students out of whom a committee of 2 professors and 3 students is to be formed. Find the number of ways in which this can be done. Further find in how many of these committees:
a particular student is included.
A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has (i) no girl?
A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has (ii) at least one boy and one girl?
A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has(iii) at least 3 girls?
A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of: atmost 3 girls?
Out of 18 points in a plane, no three are in the same straight line except five points which are collinear. How many (i) straight lines
If 20Cr = 20Cr−10, then 18Cr is equal to
If 20Cr + 1 = 20Cr − 1 , then r is equal to
5C1 + 5C2 + 5C3 + 5C4 +5C5 is equal to
How many different committees of 5 can be formed from 6 men and 4 women on which exact 3 men and 2 women serve?
(a) 6
(b) 20
(c) 60
(d) 120
If n + 1C3 = 2 · nC2 , then n =
Find n if `""^6"P"_2 = "n" ""^6"C"_2`
Find n if `""^(2"n")"C"_3: ""^"n""C"_2` = 52:3
Find n and r if `""^"n""P"_"r"` = 720 and `""^"n""C"_("n" - "r")` = 120
Four parallel lines intersect another set of five parallel lines. Find the number of distinct parallelograms that can be formed.
Find the value of 80C2
Find the value of 20C16 – 19C16
We wish to select 6 persons from 8, but if the person A is chosen, then B must be chosen. In how many ways can selections be made?
A box contains two white, three black and four red balls. In how many ways can three balls be drawn from the box, if atleast one black ball is to be included in the draw
If nCr – 1 = 36, nCr = 84 and nCr + 1 = 126, then find rC2.
If 20 lines are drawn in a plane such that no two of them are parallel and no three are concurrent, in how many points will they intersect each other?
A bag contains six white marbles and five red marbles. Find the number of ways in which four marbles can be drawn from the bag if two must be white and two red
In how many ways can a football team of 11 players be selected from 16 players? How many of them will exclude 2 particular players?
If nC12 = nC8, then n is equal to ______.
The number of ways in which a team of eleven players can be selected from 22 players always including 2 of them and excluding 4 of them is ______.
Given 5 different green dyes, four different blue dyes and three different red dyes, the number of combinations of dyes which can be chosen taking at least one green and one blue dye is ______.
There are 12 persons seated in a line. Number of ways in which 3 persons can be selected such that atleast two of them are consecutive, is ______.
A regular polygon has 20 sides. The number of triangles that can be drawn by using the vertices but not using the sides is
