PERMUTATION AND COMBINATION

Q: What is Fundamental theorem permutation and combination?

Fundamental theorem is Permutation and combination is suppose m ways of doing things and each m way are associated with n ways of doing things, so the total number of ways of doing both things is the multiplication of m and n (m × n).

Q: What is mean by Combination?

The combination means selection (or forming a group).

For permutation and combination basic concepts here,

Table of Contents

Fundamental Theorem

Suppose m ways of doing things and each m way are associated with n ways of doing things, so the total number of ways of doing both things is the multiplication of m and n (m × n).

Permutation

Permutation means arrangements.

The number of permutations of n dissimilar things taken r at a time given by,

\[\displaystyle {}^{n}{{p}_{r}}=\frac{{n!}}{{\left( {n-r} \right)!}}\]

\[\displaystyle =n(n-1)(n-2)......(n-r+1)----(1)\]

N dissimilar things, the number of permutations taken all at a time is,

\[\displaystyle {}^{n}{{p}_{n}}=n!\]

The number of circular permutations of n different things taken all at a time is (n-1)!

The number of permutations of n things, taken all at a time, when P₁ is alike of one kind, P₂ is like 2^nd kind,…..Pr is r^th kind, is given by,

\[\displaystyle \frac{{n!}}{{({{p}_{1}}!)({{p}_{2}}!)......({{p}_{r}}!)}}-----(2)\]

Combination

The combination means selection (or forming a group)

The number of combinations of n different things, taken r at a time, is given by,

\[\displaystyle {{n}_{{{{c}_{r}}}}}=\frac{{n!}}{{\left( {n-r} \right)!(r)!}}=\frac{{{}^{n}{{p}_{r}}}}{{r!}}---(3)\]

\[\displaystyle {{n}_{{{{c}_{0}}}}}=1,{}^{n}{{c}_{n}}=1----(4)\]

\[\displaystyle {{n}_{{{{c}_{p}}}}}={{n}_{{{{c}_{q}}}}}\to P+q=n\text{ or }P=q----(5)\]

\[\displaystyle {{n}_{{{{c}_{r}}}}}={{n}_{{{{c}_{{n-r}}}}}}-----(6)\]

\[\displaystyle {{n}_{{{{c}_{{r-1}}}}}}{}^{n}{{c}_{r}}={}^{{n+1}}{{c}_{r}}----(7)\]

N number of different things combinations, taken r at a time, when p particular things always given by,

\[\displaystyle ={}^{{n-p}}{{c}_{{r-p}}}----(8)\]

Number combination of n different things taken r at a time P particular things never occur is given by,

\[\displaystyle {}^{{n-p}}{{c}_{r}}---(9)\]

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PERMUTATION AND COMBINATION

Fundamental Theorem

Permutation

Combination

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Bhoomika Sheladiya

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