JEE Main & Advanced Mathematics Permutations and Combinations Number of Combinations With Repetition and All Possible Selections

(1) The number of combinations of \[n\] distinct objects taken \[r\] at a time when any object may be repeated any number of times.

= Coefficient of \[{{x}^{r}}\] in \[{{(1+x+{{x}^{2}}+.......+{{x}^{r}})}^{n}}\]

= Coefficient of \[{{x}^{r}}\] in \[{{(1-x)}^{-n}}{{=}^{n+r-1}}{{C}_{r}}\]

(2) The total number of ways in which it is possible to form groups by taking some or all of \[n\] things at a time is \[^{n}{{C}_{1}}+{{\,}^{n}}{{C}_{2}}+........+{{\,}^{n}}{{C}_{n}}={{2}^{n}}-1\].

(3) The total number of ways in which it is possible to make groups by taking some or all out of \[n=({{n}_{1}}+{{n}_{2}}+....)\] things, when \[{{n}_{1}}\] are alike of one kind, \[{{n}_{2}}\] are alike of second kind, and so on is \[\{({{n}_{1}}+1)\,({{n}_{2}}+1)......\}-1\].

(4) The number of selections of \[r\] objects out of \[n\] identical objects is 1.

(5) Total number of selections of zero or more objects from \[n\] identical objects is \[n+1\].

(6) The number of selections taking at least one out of \[{{a}_{1}}+{{a}_{2}}+{{a}_{3}}+......+{{a}_{n}}\]+ k objects, where \[{{a}_{1}}\] are alike (of one kind), \[{{a}_{2}}\] are alike (of second kind) and so on......\[{{a}_{n}}\] are alike (of n^th kind) and k are distinct

\[=[({{a}_{1}}+1)\,({{a}_{2}}+1)\,({{a}_{3}}+1).......({{a}_{n}}+1)]\,{{2}^{k}}-1\].