Advertisements
Advertisements
प्रश्न
If α = mC2, then find the value of αC2.
Advertisements
उत्तर
\[{}^\alpha C_2 = \frac{\alpha}{2} \times \frac{(\alpha - 1)}{1} \times^\alpha C_0\] [∵\[{}^n C_r = \frac{n}{r} . {}^{n - 1} C_{r - 1}\]]
\[ = \frac{1}{2} \left[ \frac{m!}{2! \left( m - 2 \right)!} \left( \frac{m!}{2! \left( m - 2 \right)!} - 1 \right) \right]\]
\[ = \frac{1}{2} \left[ \frac{m \left( m - 1 \right)}{2} \left( \frac{m \left( m - 1 \right)}{2} - 1 \right) \right]\]
\[ = \frac{1}{2} \left[ \frac{m\left( m - 1 \right)}{2} \left( \frac{m \left( m - 1 \right) - 2}{2} \right) \right]\]
\[ = \frac{1}{8} \left[ m \left( m - 1 \right) \left\{ m \left( m - 1 \right) - 2 \right\} \right]\]
\[ = \frac{1}{8} \left[ m^2 \left( m - 1 \right)^2 - 2m \left( m - 1 \right) \right]\]
\[ = \frac{1}{8} \left[ m^2 \left( m^2 + 1 - 2m \right) - 2 m^2 + 2m \right]\]
\[ = \frac{1}{8} \left[ m^4 + m^2 - 2 m^3 - 2 m^2 + 2m \right]\]
\[ = \frac{1}{8} \left[ m^4 - 2 m^3 - m^2 + 2m \right]\]
\[ = \frac{1}{8} \left[ \left( m^2 - 2m \right) \left( m^2 - 1 \right) \right]\]
\[ = \frac{1}{8} \left[ m \left( m - 2 \right) \left( m - 1 \right) \left( m + 1 \right) \right]\]
\[ = \frac{1}{8} \left( m + 1 \right) m \left( m - 1 \right) \left( m - 2 \right)\]
APPEARS IN
संबंधित प्रश्न
If nC8 = nC2, find nC2.
In how many ways can one select a cricket team of eleven from 17 players in which only 5 players can bowl if each cricket team of 11 must include exactly 4 bowlers?
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?
Compute:
L.C.M. (6!, 7!, 8!)
There are four parcels and five post-offices. In how many different ways can the parcels be sent by registered post?
Evaluate the following:
12C10
Evaluate the following:
If 15Cr : 15Cr − 1 = 11 : 5, find r.
If nC4 , nC5 and nC6 are in A.P., then find n.
How many different boat parties of 8, consisting of 5 boys and 3 girls, can be made from 25 boys and 10 girls?
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?
How many different selections of 4 books can be made from 10 different books, if
there is no restriction;
From 4 officers and 8 jawans in how many ways can 6 be chosen (i) to include exactly one officer
A candidate is required to answer 7 questions out of 12 questions which are divided into two groups, each containing 6 questions. He is not permitted to attempt more than 5 questions from either group. In how many ways can he choose the 7 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 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?
In how many ways can a team of 3 boys and 3 girls be selected from 5 boys and 4 girls?
A bag contains 5 black and 6 red balls. Determine the number of ways in which 2 black and 3 red balls can be selected.
If 20Cr = 20Cr + 4 , then rC3 is equal to
If mC1 = nC2 , then
If nCr + nCr + 1 = n + 1Cx , then x =
Three persons enter a railway compartment. If there are 5 seats vacant, in how many ways can they take these seats?
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?
The number of diagonals that can be drawn by joining the vertices of an octagon is
The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines is
Five students are selected from 11. How many ways can these students be selected if two specified students are not selected?
A student has to answer 10 questions, choosing atleast 4 from each of Parts A and 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 convex polygon has 44 diagonals. Find the number of its sides.
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 three girls.
A committee of 6 is to be chosen from 10 men and 7 women so as to contain atleast 3 men and 2 women. In how many different ways can this be done if two particular women refuse to serve on the same committee ______.
There are 12 points in a plane of which 5 points are collinear, then the number of lines obtained by joining these points in pairs is 12C2 – 5C2.
There are 10 professors and 20 lecturers out of whom a committee of 2 professors and 3 lecturer is to be formed. Find:
| C1 | C2 |
| (a) In how many ways committee: can be formed | (i) 10C2 × 19C3 |
| (b) In how many ways a particular: professor is included | (ii) 10C2 × 19C2 |
| (c) In how many ways a particular: lecturer is included | (iii) 9C1 × 20C3 |
| (d) In how many ways a particular: lecturer is excluded | (iv) 10C2 × 20C3 |
There are 15 players in a cricket team, out of which 6 are bowlers, 7 are batsmen and 2 are wicketkeepers. The number of ways, a team of 11 players be selected from them so as to include at least 4 bowlers, 5 batsmen and 1 wicketkeeper, is ______.
A badminton club has 10 couples as members. They meet to organise a mixed double match. If each wife refers to p artner as well as oppose her husband in the match, then the number of different ways can the match off will be ______.
The no. of different ways, the letters of the word KUMARI can be placed in the 8 boxes of the given figure so that no row remains empty will be ______.
The number of numbers between 2,000 and 5,000 that can be formed with the digits 0, 1, 2, 3, 4, (repetition of digits is not allowed) and are multiple of 3 is?
The number of words, with or without meaning, that can be formed by taking 4 letters at a time from the letters of the word 'SYLLABUS' such that two letters are distinct and two letters are alike is ______.
