Advertisements
Advertisements
Question
The straight lines l1, l2 and l3 are parallel and lie in the same plane. A total numbers of m points are taken on l1; n points on l2, k points on l3. The maximum number of triangles formed with vertices at these points are ______.
Options
`""^((m + n + k))"C"_3`
`""^((m + n + k))"C"_3 - ""^n"C"_3 - ""^6"C"_3 - ""^k"C"_3`
mC3 + nC3 + kC3
mC3 × nC3 × kC3
Advertisements
Solution
The straight lines l1, l2 and l3 are parallel and lie in the same plane. A total numbers of m points are taken on l1; n points on l2, k points on l3. The maximum number of triangles formed with vertices at these points are `""^((m + n + k))"C"_3 - ""^n"C"_3 - ""^6"C"_3 - ""^k"C"_3`.
Explanation:
Here the total number of points are (m + n + k) which must give `""^((m + n + k))"C"_3` number of triangles but m points on l1 taking 3 points at a time gives mC3 combinations which produce no triangle.
Similarly, nC3 and kC3 number of triangles can not be formed.
Therefore, the required number of triangles is `""^((m + n + k))"C"_3 - ""^n"C"_3 - ""^6"C"_3 - ""^k"C"_3`.
APPEARS IN
RELATED QUESTIONS
In how many ways can a team of 3 boys and 3 girls be selected from 5 boys and 4 girls?
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?
The English alphabet has 5 vowels and 21 consonants. How many words with two different vowels and 2 different consonants can be formed from the alphabet?
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?
Prove that
A mint prepares metallic calendars specifying months, dates and days in the form of monthly sheets (one plate for each month). How many types of calendars should it prepare to serve for all the possibilities in future years?
How many A.P.'s with 10 terms are there whose first term is in the set {1, 2, 3} and whose common difference is in the set {1, 2, 3, 4, 5}?
In how many ways can six persons be seated in a row?
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?
A number lock on a suitcase has 3 wheels each labelled with ten digits 0 to 9. If opening of the lock is a particular sequence of three digits with no repeats, how many such sequences will be possible? Also, find the number of unsuccessful attempts to open the lock.
Evaluate the following:
12C10
Evaluate the following:
If 28C2r : 24C2r − 4 = 225 : 11, find r.
How many different boat parties of 8, consisting of 5 boys and 3 girls, can be made from 25 boys and 10 girls?
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 triangles can be obtained by joining 12 points, five of which are collinear?
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?
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.
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: exactly 3 girls?
How many different words, each containing 2 vowels and 3 consonants can be formed with 5 vowels and 17 consonants?
Find the number of ways in which : (a) a selection
Find the number of combinations and permutations of 4 letters taken from the word 'EXAMINATION'.
If 20Cr + 1 = 20Cr − 1 , then r is equal to
If nC12 = nC8 , then n =
If 43Cr − 6 = 43C3r + 1 , then the value of r is
There are 3 wicketkeepers and 5 bowlers among 22 cricket players. A team of 11 players is to be selected so that there is exactly one wicketkeeper and at least 4 bowlers in the team. How many different teams can be formed?
Find the value of 80C2
There are 10 lamps in a hall. Each one of them can be switched on independently. Find the number of ways in which the hall can be illuminated.
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 no girls
15C8 + 15C9 – 15C6 – 15C7 = ______.
In a football championship, 153 matches were played, Every two teams played one match with each other. The number of teams, participating in the championship is ______.
If some or all of n objects are taken at a time, the number of combinations is 2n – 1.
Number of selections of at least one letter from the letters of MATHEMATICS, is ______.
Total number of 6-digit numbers in which only and all the five digits 1, 3, 5, 7 and 9 appear is ______.
From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on the shelf so that the dictionary is always in the middle. Then, the number of such arrangements is ______.
