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- If certain objects are to be arranged in such a way that the order of objects is not important, then the concept of combinations is used.
- The number of combinations of n things taken r (0 < r < n) at a time is given by
^{n}C_{r}= n!/r!(n-r)! - The relationship between combinations and permutations is
^{ n}C_{r}=^{n}P_{r}/r! - The number of ways of selecting r objects from n different objects subject to certain condition like:

1. k particular objects are always included =^{ n-k}C_{r-k}

2. k particular objects are never included =^{ n-k}C_{r} - The number of arrangement of n distinct objects taken r at a time so that k particular objects are

1. Always included =^{n-k}C_{r-k}.r!,

2. Never included =^{n-k}C_{r}.r!. - In order to compute the combination of n distinct items taken r at a time wherein, the chances of occurrence of any item are not fixed and may be one, twice, thrice, …. up to r times is given by
^{n+r-1}C_{r} - If there are m men and n women (m > n) and they have to be seated or accommodated in a row in such a way that no two women sit together then total no. of such arrangements

=^{m+1}C_{n}. m! This is also termed as the Gap Method. - If there is a problem that requires n number of persons to be accommodated in such a way that a fixed number say ‘p’ are always together, then that particular set of p persons should be treated as one person. Hence, the total number of people in such a case becomes (n-m+1). Therefore, the total number of possible arrangements is (n-m+1)! m! This is also termed as the String Method.
- Let there be n types of objects with each type containing at least r objects. Then the number of ways of arranging r objects in a row is nr.
- The number of selections from n different objects, taking at least one

=^{n}C_{1}+^{n}C_{2}+^{n}C_{3 }+ ... +^{n}C_{n}= 2^{n}- 1. - Total number of selections of zero or more objects from n identical objects is n+1.
- Selection when both identical and distinct objects are present:
- The number of selections, taking at least one out of a
_{1}+ a_{2}+ a_{3}+ ... a_{n}+ k objects, where a1 are alike (of one kind), a_{2}are alike (of second kind) and so on ... an are alike (of nth kind), and k are distinct

= {[(a_{1}+ 1)(a_{2}+ 1)(a_{3}+ 1) ... (a_{n}+ 1)]2^{k}} - 1. - Combination of n different things taken some or all of n things at a time is given by 2
^{n }– 1. - Combination of n things taken some or all at a time when p of the things are alike of one kind, q of the things are alike and of another kind and r of the things are alike of a third kind

= [(p + 1) (q + 1)(r + 1)….] – 1 - Combination of selecting s
_{1}things from a set of n_{1}objects and s_{2 }things from a set of n_{2 }objects where combination of s1 things and s2 things are independent is given by^{ n1}C_{s1}x^{n2}C_{s2 } - Some results related to
^{n}C_{r }

1.^{n}C_{r }=^{ n}C_{n-r}

2. If^{n}C_{r }=^{n}C_{k}, then r = k or n-r = k

3.^{n}C_{r }+^{n}C_{r-1}=^{n+1}C_{r}

4.^{n}C_{r}= n/r^{n-1}C_{r-1}

5.^{n}C_{r}/^{n}C_{r-1}= (n-r+1)/ r

6. If n is even^{n}C_{r}is greatest for r = n/2

7. If n is odd, is greatest for r = (n-1) /2, (n+1)/2

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