Question

Tutorial on Permutation and Combination. Part 2

Answer 1

1. ‘r’ things can be selected out of ‘n’ identical things in only one way.

2. if \[\huge ^nP_r \] denotes the number of permutation of ‘n’ different things, taking ‘r’ at a time, then
\[^{ n }P_{ r }=n(n-1)(n-2).......(n-r+1)=\frac { n! }{ (n-r)! } \]

3. Note that \[^{ n }P_{ n } = n! \]

4. If \[^nC_r\] denotes the number of combination of ‘n’ different things, taking ‘r’ @ a time then
\[ ^{ n }C_{ r }=\frac { n! }{ r!(n-r)! } =\frac { ^{ n }P_{ r } }{ r! } \quad where\quad r\le \quad n;\quad n\quad \in\quad N\quad \& r\quad \in\quad W\]

5. The number of circular Permutation of ‘n’ different things taken all at a time is (n-1)! . If clockwise and anticlockwise circular permutation are considered to be same, then its \[\frac { (n-1)! }{ 2 } \]
Remark : Number of circular permutation of ‘n’ things when ‘p’ alike and the rest different taken all at a time distinguishing clockwise and anticlockwise arrangement is \[\frac { (n-1)! }{ p! } \]

6. ‘n’ persons can sit around a circular table in (n-1)! Ways.

7. Given ‘n’ different objects, the number of ways of selecting at least one of them is,
^nC_1+^nC_2 +^nC_3+.......+^nC_N = 2^n-1.
This can also be stated as the total number of combination of ‘n’ distinct things.

8. Total number of ways in which it is possible to make a selection by taking some or all out of ‘p+q+r+…….. things, where ‘p’ are alike one kind, ‘q’ alike of the second kind & so on is given by : ‘(p+1)(q+1)(r+!)………-1’

9. Number of ways in which it is possible to make a selection of ‘m+p+n=N’ things, where ‘p’ are alike one kind ‘m’ alike the second kind & ‘n’ alike the third kind taken ‘r’ at a time given coefficient of x^r

10. Number of ways in which ‘n’ identical things may be distributed among ‘p’ persons if each person may receive none, one or more thing is; ^{ n+p-1 }C_n
\[^{ n }C_{ r }=^{ n }C_{ n-1\quad };\quad ^nC_0 = \ ^nC_n=1 \]
\[^{ n }C_{ x }=^{ n }C_{ y\quad }\rightarrow \quad x=y\quad or\quad x+y=\quad n\]
\[^{ n }C_{ r }=^{ n }C_{ r-1\quad }=^{ n+1 }C_{ r } \]

11.\[ ^{ n }C_{ r } \] is maximum if (a) r=n/2, if n is even.
(b) r=(n-1)/2 if n is odd

Answer 2

thanks Lg

Answer 3

@mathslover have a look

Answer 4

@ParthKohli have a look

Answer 5

Congratulations. http://www.math-for-all-grades.com/pandc.html

Answer 6

then!

Answer 7

here's my work

Answer 8

'Your' work?

Answer 9

yes!

Answer 10

Last 2 hrs I was working with the write and LaTeX

Answer 11

http://www.math-for-all-grades.com/pandc.html See 'your' work here.
Well, it's nice that you haven't *exactly* copied and pasted it.

Answer 12

Man Every where the Concept is same

Answer 13

right

Answer 14

But the words are not exactly the same. Well, it's okay.

Answer 15

If any doubt U can call my Sir . Dr. BK Parghania

Answer 16

I go to his Coaching

Answer 17

you took it bck

Answer 18

I thought that you didn't want it bc you deleted that reply

Answer 19

Sorry !!

Answer 20

@ParthKohli @lgbasallote there's a user when I click it I directly go to home! why ??