Koundinya-Mathematics Teachers Hub: A note on Permutations and Combinations

Here is a situation where one answer has many questions.
${\color{blue} 1) (n+r-1)C_{r-1}}$
${\color{blue} 2) (n+r-1)C_{n}}$
${\color{blue} 3) No \ {}of\ {} terms \ {}in\ {}the\ {} expansion\ {}of\ {}the\ {}multinomial \\ (x_{1}+x_{2}+x_{3}+...x_{r})^n}$
${\color{Yellow} 4) No \ {}of\ {} nonnegative \ {}integral\ {}solutions\ {} of\ {}the\ {}equation\ x_{1}+x_{2}+x_{3}+...x_{r}=n}$
${\color{blue} 5) \ {}No \ {}of\ {} Permutations \ {}of\ {}n\ {} identical\ {}things\ {}of\ {}one \ {}kind\ {}and\ {}r-1\ {}identical\ {}things}$
${\color{blue} \ {} of\ {} another \ {}kind\ {}taken\ {}all\ {}at \ {}a \ {}time}$
${\color{blue} 6) \ {}No \ {}of\ {} ways \ {}of\ {}dividing\ {} n+r-1\ {}things\ {}into\ {}two \ {}groups\ {}of\ {}\\ n\ {}and\ {}r-1\ {} each}$
${\color{blue} 7) \ {}Coefficient \ {}of\ {} x^{n} \ {}in\ (1-x)^{-r}}$
${\color{Yellow} 8) \ {}Coefficient \ {}of\ {} x^{n} \ {}in\ (1+x+x^{2}+...x^{n})^{r}}$
${\color{blue} 9) \ {}Number \ {}of\ {}ways\ {}of\ {}placing\ {}n\ {}identical\ {}things\\into\ {}r\ {}boxes\ {}where\ {}boxes\ {}are\ {}allowed}$
${\color{blue} \ {}to\ {}be\ {}empty\ {}and\ {}each\ {}box\ {}is\ {}capable\ {}of\ {}holding\ {}all\ {}n\ {}identical\ {}things}$

...Try to add to the list!!!...are these independent of each other?

Koundinya-Mathematics Teachers Hub

Tuesday, 27 December 2011

A note on Permutations and Combinations

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