PERMUTATION AND COMBINATIONS
Permutation : It is defined as the ways of arranging
The number of permutations of n objects taken r at a time is
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n = n (n-1) (n-2) (n-3)…(n-r+1)
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n! n
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=———– nP
= n! nP
= —- = 1
(n – r)! n
The factorial
The product of n natural numbers 1,2,3…n is denoted as n! This we read as factors n. Some results are
o! = 1
n! = n(n -1) (n – 2)
Some results of permutation :
- The total number of permutation of n items all together, when p items are of one type, Q are of second type and r of then third kind and the
n! remaining are of different type is ———
p!q!r!
- The number of permutations of a different items teken r at a time, when each may be repeated any number of
n(nr – 1) times is ———–
n – 1
- If there are n objects of which m are like and the remaining (n-m) are differ- ent, then the total number of
n!
permutations are———— –. This is called permutations of like objects.
m!(n – m)!
- The number of circular permutations of n different objects is n-1! because if there are n! linear permutations there
n!
will be—– = (n-1)! circular permutations.
n
Combinations :
When r objects taken out of n objects then combination of n objects taken r at a time, we write
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n! n(n – 1)…(n – r + 1) nP
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n = C (n,r) = = =
r!(n -r)! r! r!
- Number of combinations of a different objects taken n at a time
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n! nC n – r + 1 nC n
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n = 1 Similarly nC = 1 = = = =
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n!(n – n)! nC
– 1 r n – 1 C
r – 1 r
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- Value of nCl = nCn
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- The sum of nC
n
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r-1
= n+1C
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(4) nC
= nC
Þ p = q or p+q=n
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n 1 1
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(5) The ratio of = —— = —- = —–
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- nC
- nC
n
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r+1
= nC
n
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r
= n+1C
r! rP
r r+1 r
- The total number of ways by which to divide n identical objects among r
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people when each gets at least one is n-qC
- The number of ways in which m+n objects can be divided in to two groups m + n
having m and n objects = ——–
m! n!
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- The total number of diagouals in a polygon of n sides is nC –
