From a total of 9 players a basketball team of playing 5 is to be selected. How many teams are possible if (i) the distinct positions of the playing 5 are to be taken into consideration. (ii) the distinct position of the playing 5 are not taken into consideration. (iii) the disti

Question

From a total of 9 players a basketball team of playing 5 is to be selected. How many teams are possible if (i) the distinct positions of the playing 5 are to be taken into consideration. (ii) the distinct position of the playing 5 are not taken into consideration. (iii) the distinct positions are not taken into consideration, but two players either Krish or Rohit (but not both) should be in the playing 5.

Accepted Answer

> [!TLDR] Choosing 5 players out of 9 for a basketball team, considering three scenarios: positions distinguished (arrangement), positions not distinguished (selection), and a selection restricted to exactly one of two named players.

> [!FORMULA] When the 5 chosen players are also assigned to distinct labelled positions, order matters, so we use permutations:
> $$^{n}P_r = \dfrac{n!}{(n-r)!}$$
> When only the group membership matters (no position labels), order doesn't matter, so we use combinations:
> $$^{n}C_r = \dfrac{n!}{r!\,(n-r)!}$$
> Here $n=9$ (total players), $r=5$ (team size).

**(i) Distinct positions considered (arrangement)**

1. Since each of the 5 slots is a different labelled position, the same 5 players in a different arrangement count as a different team.
2. Number of ways $= {}^{9}P_5 = \dfrac{9!}{(9-5)!} = \dfrac{9!}{4!} = 9	imes8	imes7	imes6	imes5$.
3. Compute: $9	imes8=72$, $72	imes7=504$, $504	imes6=3024$, $3024	imes5=15120$.

**(ii) Distinct positions NOT considered (pure selection)**

4. Now only WHICH 5 players are chosen matters, not their arrangement — combinations.
5. Number of ways $= {}^{9}C_5 = \dfrac{9!}{5!\,4!}$.
6. Compute: $\dfrac{9	imes8	imes7	imes6}{4	imes3	imes2	imes1} = \dfrac{3024}{24} = 126$.

**(iii) No positions, but exactly one of Krish or Rohit (not both) must play**

7. "Exactly one of the two" means: choose which one of {Krish, Rohit} is on the team — $2$ ways (Krish-in-Rohit-out, or Rohit-in-Krish-out). …

Q.From a total of 9 players a basketball team of playing 5 is to be selected. How many teams are possible if

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Q.From a total of 9 players a basketball team of playing 5 is to be selected. How many teams are possible if

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