Conditional Combinations
Category : JEE Main & Advanced
(1) The number of ways in which r objects can be selected from n different objects if k particular objects are
(i) Always included = \[^{n-k}{{C}_{r-k}}\]
(ii) Never included = \[^{n-k}{{C}_{r}}\]
(2) The number of combinations of n objects, of which p are identical, taken r at a time is
\[^{n-p}{{C}_{r}}{{+}^{n-p}}{{C}_{r-1}}{{+}^{n-p}}{{C}_{r-2}}+.......{{+}^{n-p}}{{C}_{0}}\], if \[r\le p\] and
\[^{n-p}{{C}_{r}}{{+}^{n-p}}{{C}_{r-1}}{{+}^{n-p}}{{C}_{r-2}}+.......{{+}^{n-p}}{{C}_{r-p}}\], if \[r>p\].
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