In how many ways can a group of 10 players be formed from 14 state level players and 4 district level players such that the group contains exactly 1 district level player?

Key Concepts Needed

1. Combination (nCr):
   ${^nC_r} = \frac{n!}{r!(n-r)!}$
   It counts the number of ways to choose $r$ objects from $n$ objects when order does not matter.

2. Independent Choices Multiply: If you can make Choice-A in $m$ ways and, independently, Choice-B in $n$ ways, then the pair of choices can be made in $m \times n$ ways.

Applying to the Problem

• 14 state players ⇒ we label them $S_1, S_2, \dots, S_{14}$.
• 4 district players ⇒ we label them $D_1, D_2, D_3, D_4$.

We want a team of 10 players with exactly one district player.

1. Pick the district player
   $\text{Ways} = {^4C_1}$
2. Pick the remaining 9 state players
   $\text{Ways} = {^{14}C_9}$

Because these choices do not interfere with each other, total ways = product of the two results.

Imagine you have two kinds of marbles

• Blue marbles = 14 (these are the state players)
• Red marbles = 4 (these are the district players)

You want to pick exactly 10 marbles out of the 18, but you must get exactly one red marble.

So what do you really have to decide?

1. Which red marble (district player) you take.
2. Which nine blue marbles (state players) you take.

Once you decide those two things, your team is fixed.

Counting:

• There are 4 ways to pick one red marble (because there are 4 red marbles).
• After that, you need 9 blue marbles out of 14 blue marbles. The number of ways to do that is a bit bigger, but it’s just a combination calculation: choose 9 out of 14.

Finally, multiply those two counts together because each independent choice can go with every choice of the other. That total is your answer!

Step-by-Step Solution

1. Choose the district player:

   $$^4C_1 = 4$$

2. Choose 9 state players:

   $$^{14}C_9 = ^{14}C_{14-9} = ^{14}C_5 = \frac{14!}{5!\,9!} = 2002$$

3. Multiply the independent counts:

   $$\text{Total ways} = 4 \times 2002 = 8008$$

Final Answer: 8008 ways.