Advertisements
Advertisements
Question
The number of triangles that are formed by choosing the vertices from a set of 12 points, seven of which lie on the same line is ______.
Options
105
15
175
185
Advertisements
Solution
The number of triangles that are formed by choosing the vertices from a set of 12 points, seven of which lie on the same line is 185.
Explanation:
Total number of triangles formed from 12 points taking 3 at a time = 12C3
But given that out of 12 points, 7 are collinear
So, these seven points will form no triangle.
∴ The required number of triangles = 12C3 – 7C3
= `(12!)/(3! 9!) - (7!)/(3!4!)`
= `(12 xx 11 xx 10 xx 9!)/(3 xx 2 xx 1 xx 9!) - (7 xx 6 xx 5 xx 4!)/(3 xx 2 xx 1 xx 4!)`
= `(12 xx 11 xx 10)/(3 xx 2) - (7 xx 6 xx 5)/(3 xx 2)`
= 220 – 35
= 185
APPEARS IN
RELATED QUESTIONS
If nC8 = nC2, find nC2.
How many 6-digit numbers can be formed from the digits 0, 1, 3, 5, 7 and 9 which are divisible by 10 and no digit is repeated?
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?
Prove that
There are four parcels and five post-offices. In how many different ways can the parcels be sent by registered post?
Twelve students complete in a race. In how many ways first three prizes be given?
How many different five-digit number licence plates can be made if
the first-digit cannot be zero, but the repetition of digits is allowed?
Serial numbers for an item produced in a factory are to be made using two letters followed by four digits (0 to 9). If the letters are to be taken from six letters of English alphabet without repetition and the digits are also not repeated in a serial number, how many serial numbers are possible?
Evaluate the following:
35C35
Evaluate the following:
If 15Cr : 15Cr − 1 = 11 : 5, find r.
If 28C2r : 24C2r − 4 = 225 : 11, find r.
If 16Cr = 16Cr + 2, find rC4.
In how many ways can a football team of 11 players be selected from 16 players? How many of these will
include 2 particular players?
From a class of 12 boys and 10 girls, 10 students are to be chosen for a competition; at least including 4 boys and 4 girls. The 2 girls who won the prizes last year should be included. In how many ways can the selection be made?
How many different selections of 4 books can be made from 10 different books, if
two particular books are always selected;
How many different selections of 4 books can be made from 10 different books, if two particular books are never selected?
From 4 officers and 8 jawans in how many ways can 6 be chosen (i) to include exactly one officer
A student has to answer 10 questions, choosing at least 4 from each of part A and part B. If there are 6 questions in part A and 7 in part B, in how many ways can the student choose 10 questions?
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 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: at least 3 girls?
In an examination, a question paper consists of 12 questions divided into two parts i.e., Part I and Part II, containing 5 and 7 questions, respectively. A student is required to attempt 8 questions in all, selecting at least 3 from each part. In how many ways can a student select the questions?
Find the number of ways in which : (a) a selection
A tea party is arranged for 16 persons along two sides of a long table with 8 chairs on each side. Four persons wish to sit on one particular side and two on the other side. In how many ways can they be seated?
If 20Cr + 1 = 20Cr − 1 , then r is equal to
There are 13 players of cricket, out of which 4 are bowlers. In how many ways a team of eleven be selected from them so as to include at least two bowlers?
If n + 1C3 = 2 · nC2 , then n =
Find the number of ways of dividing 20 objects in three groups of sizes 8, 7, and 5.
In a small village, there are 87 families, of which 52 families have atmost 2 children. In a rural development programme 20 families are to be chosen for assistance, of which atleast 18 families must have at most 2 children. In how many ways can the choice be made?
If nCr – 1 = 36, nCr = 84 and nCr + 1 = 126, then find rC2.
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 they can be of any colour
In how many ways can a football team of 11 players be selected from 16 players? How many of them will include 2 particular players?
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 at least one boy and one girl
Total number of words formed by 2 vowels and 3 consonants taken from 4 vowels and 5 consonants is equal to ______.
Number of selections of at least one letter from the letters of MATHEMATICS, 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 ______.
There are ten boys B1, B2, ...., B10 and five girls G1, G2, ...., G5 in a class. Then the number of ways of forming a group consisting of three boys and three girls, if both B1 and B2 together should not be the members of a group is ______.
