## Combinations

More Concepts and Formulas - CombinationsNumber of combinations of n distinct objects taking r at a time when each object may be repeated any number of times

=

=

^{(n+r-1)}C_{r}
Number of ways in which one or more objects can be selected from n distinct objects

=

*(i.e., we can select 1 or 2 or 3 or … or n objects at a time)*=

^{n}C_{1}+^{n}C_{2}+ ... +^{n}C_{n}= 2^{n}- 1
Number of ways in which one or more objects can be selected out of S

= (S

The above formula can be generalized as follows.

Number of ways in which one or more objects can be selected out of S

= (S

_{1}alike objects of one kind, S_{2}alike objects of second kind and S_{3}alike objects of third kind= (S

_{1}+ 1)(S_{2}+ 1)(S_{3}+ 1) - 1The above formula can be generalized as follows.

Number of ways in which one or more objects can be selected out of S

_{1}alike objects of one kind, S_{2}alike objects of second kind , S_{3}alike objects of third kind and so on ... S_{n}alike objects of n^{th}kind= (S

_{1}+ 1) (S_{2}+ 1)(S_{3}+ 1)...(S_{n}+ 1) - 1
Number of ways in which one or more objects can be selected out of S

= (S

The above formula can be generalized as follows.

Number of ways in which one or more objects can be selected out of S

= (S

_{1}alike objects of one kind, S_{2}alike objects of second kind and rest p different objects= (S

_{1}+ 1)(S_{2}+ 1)2^{p}- 1The above formula can be generalized as follows.

Number of ways in which one or more objects can be selected out of S

_{1}alike objects of one kind, S_{2}alike objects of second kind and so on ... S_{n}alike objects of n^{th}kind and rest p different objects= (S

_{1}+ 1) (S_{2}+ 1) ... (S_{n}+ 1) 2^{p}- 1
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