Consider following examples,
ex 1: Selection which can be made from the letters a,b,c by taking 2 at a time are ab,bc,ca.
ex 2: Selection which can be made from the letters a,b,c taken all at a time is abc.
ex 3: Groups which can be made from the letters a,b,c,d by taking 3 at a time are abc,abd,acd,bcd
Here, the order is not important.
Each of the groups or selections which can be made by taking some or all of a number of objects without reference to the order of objects in each group is called a combination.
Each selection of ex 1 is called a combination of 3 different objects taken 2 at a time.
Each selection of ex 2 is called a combination of 3 different objects taken 1 at a time.
Each selection of ex 3 is called a combination of 4 different objects taken 3 at a time.
Let $r$ and $n$ be two positive integers and $0\leq r\leq n$, then the number of combinations of $n$ different objects taken $r$ at time is denoted by $^{n}C_{r}$
Combinations when all objects are different
Number of combinations of $n$ different objects taken $r$ at a time is $^{n}C_{r}$ .
Take any one of these combinations. It contains $r$ objects which can be arranged among themselves in $r!$ ways.
Thus one combination gives rise to $r!$ permutations.
Therefore $^{n}C_{r}$ combinations will give rise to $^{n}C_{r}\times r!$ permutations.
But we know that the number of permutations of $n$ different objects taken $r$ at time is $^{n}P_{r}$ .
∴ $^{n}C_{r}\times r!=^{n}P_{r}$
$^{n}C_{r}=\frac{^{n}P_{r}}{r!}$
$^{n}C_{r}=\frac{n!}{r!(n-r)!}$
Note: (i) $^{n}C_{n}=1$
(ii) $^{n}C_{0}=1$
(iii) $^{n}C_{r}=^{n}C_{n-r}$
(iv) $^{n}C_{a}=^{n}C_{b}$ ⇒ a=b or n=a+b
(v) $^{n}C_{r}+^{n}C_{r-1}=^{n+1}C_{r}$
ex 1: Selection which can be made from the letters a,b,c by taking 2 at a time are ab,bc,ca.
ex 2: Selection which can be made from the letters a,b,c taken all at a time is abc.
ex 3: Groups which can be made from the letters a,b,c,d by taking 3 at a time are abc,abd,acd,bcd
Here, the order is not important.
Each of the groups or selections which can be made by taking some or all of a number of objects without reference to the order of objects in each group is called a combination.
Each selection of ex 1 is called a combination of 3 different objects taken 2 at a time.
Each selection of ex 2 is called a combination of 3 different objects taken 1 at a time.
Each selection of ex 3 is called a combination of 4 different objects taken 3 at a time.
Let $r$ and $n$ be two positive integers and $0\leq r\leq n$, then the number of combinations of $n$ different objects taken $r$ at time is denoted by $^{n}C_{r}$
Combinations when all objects are different
Number of combinations of $n$ different objects taken $r$ at a time is $^{n}C_{r}$ .
Take any one of these combinations. It contains $r$ objects which can be arranged among themselves in $r!$ ways.
Thus one combination gives rise to $r!$ permutations.
Therefore $^{n}C_{r}$ combinations will give rise to $^{n}C_{r}\times r!$ permutations.
But we know that the number of permutations of $n$ different objects taken $r$ at time is $^{n}P_{r}$ .
∴ $^{n}C_{r}\times r!=^{n}P_{r}$
$^{n}C_{r}=\frac{^{n}P_{r}}{r!}$
$^{n}C_{r}=\frac{n!}{r!(n-r)!}$
Note: (i) $^{n}C_{n}=1$
(ii) $^{n}C_{0}=1$
(iii) $^{n}C_{r}=^{n}C_{n-r}$
(iv) $^{n}C_{a}=^{n}C_{b}$ ⇒ a=b or n=a+b
(v) $^{n}C_{r}+^{n}C_{r-1}=^{n+1}C_{r}$
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