Discrete Mathematics: Combinations Without Repetition

Combinations Without Repetition


  1. How many teams of $L$ people can be formed from $N$ people?

    2. Example. $N=4, L=0,1,2,3,4$


    $$\boxed{\binom{N}{L}}$$
  2. How many teams of $L$ people can be formed from $N$ people if a specific person $A$ must be on the team?

    3. Example. $N=4, L=1,2,3,4$


    $$\boxed{\binom{N-1}{L-1}}$$
  3. How many teams of $L$ people can be formed from $N$ people if a specific person $A$ cannot be on the team?

    4. Example. $N=4, L=0,1,2,3$


    $$\boxed{\binom{N-1}{L}}$$
  4. How many teams of $L$ people can be formed from $N$ people if a set of $K$ specific people must be on the team?

    5. Example. $N=4, L=2,3,4, K=2, \{A,B\}$


    $$\boxed{ \binom{N-K}{L-K} }$$
  5. How many teams of $L$ people can be formed from $N$ people if all $K$ specific people must not be on the team?

    6. Example. $N=4, L=2,3,4, K=2, \{A,B\}$


    $$\boxed{ \binom{N-K}{L}}$$
  6. How many teams of $L$ people can be formed from $N$ people if at least one of $K$ specific people is on the team?

    7. Example. $N=4, L=0,1,2,3,4, K=2, \{A,B\}$


    \[ \boxed{ \binom{N}{L} - \binom{N-K}{L} } \]
  7. How many teams of $L$ people can be formed from $N$ people if $K$ people stick together?

    Hint: Consider the cases where all are on the team and where none are on the team.

    8. Example. $N=4, L=1,2,3,4, K=2, \{A,B\}$


    \[ \boxed{ \binom{N-K}{L} + \binom{N-K}{L-K} } \]
  8. How many teams of $L$ people can be formed from $N$ people if no two of $K$ specific people can be on the team together? Hint: Consider the cases where exactly one of them is on the team and where none are on the team.

    9. Example. $N=4, L=0,1,2,3,4, K=2, \{A,B\}$


    \[ \boxed{ \binom{N-K}{L} + K\binom{N-K}{L-1} } \]

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