Odisha State Board CHSE Odisha Class 11 Math Notes Chapter 8 Permutations And Combinations will enable students to study smartly.
CHSE Odisha 11th Class Math Notes Chapter 8 Permutations And Combinations
Fundamental Principle Of Counting:
(a) Fundamental principle of Multiplication:
If we choose an element from set A with m element and then one element from set B with n elements, then are total number of ways we can make a choice is exactly mn.
OR
If an event can occur in m different ways, following which another event can occur in n different ways, then the total number of ways in which both the events can occur in succession in mn ways.
(b) Fundamental Principle of addition: If there are two events such that they can be performed independently in m and n different ways respectively, then either of two events can be performed in (m + n) ways.
Note:
(a) Use the multiplication principle if by doing one part of the job, the job remains incomplete.
(b) Use the addition principle if by doing one part of the job, the job is completed.
Factorial Notation:
If n ∈ N then the factorial of n, denoted by n! or ∠n is defined as
n! = n (n – 1). (n – 2) … 3.2.1.
Note:
0! = 1
Properties of Factorial:
(1) Factorial of negative integers is not defined
(2) n! = n(n – 1)!
= n(n – 1) (n – 2)!
= n(n – 1) (n – 2) (n – 3)!
(3) \(\frac{n !}{r !}\) = n(n – 1) (n – 2) ….. (r + 1)
(4) Exponent of a prime number p in n! denoted by
\(\mathrm{E}_p(n !)=\left[\frac{n}{p}\right]+\left[\frac{n}{p^2}\right]+\ldots \ldots\)
Permutation:
Each of the arrangements which can be made by taking some or all objects or things at a time is called a permutation.
(a) Permutation of n different objects:
- Number of permutations of n different objects have taken all at a time = \({ }^n \mathrm{P}_n\) = n!.
- Number of permutations of n different objects taken none at a time = \({ }^n \mathrm{P}_0\) = 1
- Number of permutations of n different objects taken r at a time = \({ }^n \mathrm{P}_r\) = P(n, r) = \(\frac{n !}{(n-r) !}\)
(b) Permutation ofnon-distinct objects:
(1) Number of permutations of n objects taken all at a time of which p objects are of same kind and others are distinct = \(\frac{n !}{p !}\)
(2) Number of permutations of n objects taken all at a time of which p objects are of one kind, q objects are of a second kind and other are distinct = \(\frac{n !}{p ! q !}\)
(3) Number of permutations of n objects taken all at a time in which p1 objects are of one kind, p2 are of second kind, p3 are 3rd kind ….. and
pn are of nth kind and other are distinct. = \(\frac{n !}{p_{1} ! p_{2} ! \cdots p_{n} !}\)
(c) Restricted permutations:
- Permutation of distinct objects with repetition: The number of permutations of n different things taken r at a time when each thing may be repeated any number of times = nr
- Number of permutations of n different things taken r at a time when a particular thing is to be always included in each arrangement = r. n-1Pr-1.
- Number of permutations of n different things, taken r at the time when p particular are to be always included in each arrangement = P(r – (p – 1) n-pPr-p.
- Number of permutations of n different things taken r at a time, when a particular thing is never taken in each arrangement = n-1Pr.
- Number of permutations of n different things taken r at a time, when p particular things never taken in each arrangement = n-pPr.
(d) Circular permutation:
(1) When we do an arrangement of objects along a closed curve we call it the circular permutation.
(2) Number of circular permutations of n distinct objects taken all at a time = (n – 1)!, where clockwise and anti-clockwise orders are taken as different, as arrangements round a table.
(3) Number of circular permutations of n distinct objects taken all at a time, where clockwise and anti-clockwise orders make no difference as beads or flowers in a necklace or garland.
= \(\frac{(n-1) !}{2}\)
(4) Number of circular permutations of n different things taken r at a time where clockwise and anti-clockwise orders are different = \(\frac{\left({ }^n \mathrm{P}_r\right)}{r}\)
(5) Number of circular permutations of n different things taken r at a time where clockwise and anti-clockwise orders make no difference = \(\frac{\left({ }^n \mathrm{P}_r\right)}{2 r}\)
(e) Some more restricted permutations:
- Number of permutations of n different things taken all at a time, when m specified things come together = m!(n – m + 1)!.
- Number of permutations of n different things taken all at a time when m specified things never come together = n! – m!(n – m + 1)!.
Combinations:
Each of the different selections made by taking some or all objects at a time irrespective of any order is called a combination.
(a) Difference between permutation and combination:
- A combination is a selection but a permutation is not a selection but an arrangement.
- In combination the order of appearance of objects is immaterial, whereas in a permutation the ordering is essential.
- Practically to find permutations of n different objects taken r at a time, we first select objects then we arrange them.
- One combination corresponds to many permutations.
(b) Combinations of n different things taken r at a time:
The number of combinations of n different things have taken r at a time ncr = C(n, r) = \(\left(\begin{array}{l}
n \\
r
\end{array}\right)=\frac{n !}{r !(n-r) !}\)
(c) Properties of ncr :
(1) ncr = nC0 = 1, nC1 = n
(2) nCr = nCn-r
(3) nCr + nCr-1 = n+1Cr (Euler’s formula)
(4) nCx = nCy ⇒ x = y or x + y = n
(5) n. n-1Cr-1 = (n – r + 1) nCr-1
(6) nCr = \(\frac{n}{r}{ }^{n-1} \mathrm{C}_{r-1}\)
(7) \(\frac{{ }^n \mathrm{C}_r}{{ }^n \mathrm{C}_{r-1}}=\frac{n-r+1}{r}\)
(8) If n is even then the greatest value of nCr is nCn/2.
⇒ If n is odd then the greatest value of nCr is \({ }^n \mathrm{C}_{\left(\frac{n+1}{2}\right)} \text { or }{ }^n \mathrm{C}_{\left(\frac{n-1}{2}\right)}\)
(d) Number of combinations of n different things taken r at a time, when k particular things always occur = n-kCr-k
(e) The number of combinations of n different things, taken r at a time where k particular things never occur = n-kCr
(f) The total number of combinations of n different things taken one or more at a time (or the number of ways of n different things selecting at least one of them) = nC1 + nC2 + nC3 + ….. + nCn = 2n -1
(g) The number of combinations of n identical things taken r at a time = 1.
(h) Number of ways of selecting r things out of n alike things where r = 0, 1, 2, 3 ….. n is (n+ 1).
(i) Division into groups:
- The number of ways in which (m + n) different things can be divided into two groups which contain m and n things respectively = \(\frac{(m+n) !}{m ! n !}\) for m ≠ n.
- If m-n then the groups are of equal size. Thus, division can be done in two ways as:
⇒ If order of groups is not important: In this case the number of ways = \(\frac{(2 n) !}{2 !(n !)^2}\)
⇒ If order of groups is important: In this case the number of ways = \(\frac{(2 n) !}{(n !)^2}\)
(j) Arrangement in groups:
- The number of ways in which n different things can be arranged into r different groups = n+r-1Pn or n! n-1Cr-1
- The number of ways in which n different things can be distributed into r different groups = rn – rC1(r – 1)n + rC2(r – 2)n ….. + (-1)r-1 . rCr-1. (Blank groups are not allowed)
- The number of ways in which n identical things can be distributed into r different groups where blank groups are allowed
= (n+r-1)C(r-1)
= (n+r-1)Cn - Number of ways in which n identical things can be distributed into r different groups where blank groups are not allowed (each group receives at least one item) = n-1Cr-1
(k) Number of divisors:
