Discrete Mathematics: Combinations With Repetition

Combinations With Repetition


  1. How many combinations are there for \(N\) dice with \(S\) sides \(1, \ldots, S\)? Hint: Let \(X_1, \ldots, X_S\) represent the number of dice with each subscripted face value.

    2. Example. $N=3, S=6, \text{Die}_1 \leq \text{Die}_2 \leq \text{Die}_3$


    \[ \boxed{\binom{N+S-1}{S-1}} \]
  2. How many combinations are there for \(N\) dice with \(S\) sides \(1, \ldots, S\) such that the face value \(1\) appears exactly \(K\) times? Hint: Let \(X_2, \ldots, X_S\) represent the number of dice with each subscripted face value.

    3. Example. $N=3, S=6, \text{Die}_1 \leq \text{Die}_2 \leq \text{Die}_3, K=1,2$


    \[ \boxed{\binom{N-K+S-2}{S-2}} \]
  3. How many combinations are there for \(N\) dice with \(S\) sides \(1, \ldots, S\) such that the face value \(1\) appears at least \(K\) times? Hint: Let \(X_1, \ldots, X_S\) represent the number of dice with each subscripted face value.

    4. Example. $N=3, S=6, \text{Die}_1 \leq \text{Die}_2 \leq \text{Die}_3, K=1,2$


    \[ \boxed{\binom{N-K+S-1}{S-1}} \]
  4. How many combinations are there for \(N\) dice with \(S\) sides \(1, \ldots, S\) such that the maximum value of the dice is \(M\)? Hint: Let \(X_1, \ldots, X_M\) represent the number of dice with each subscripted face value.

    5. Example. $N=3, S=6, \text{Die}_1 \leq \text{Die}_2 \leq \text{Die}_3, M=4$


    \[ \boxed{\binom{N+M-2}{M-1}} \]
  5. How many combinations are there for \(N\) dice with \(S\) sides \(1, \ldots, S\) such that the minimum value of the dice is \(M\)? Hint: Let \(X_M, \ldots, X_S\) represent the number of dice with each subscripted face value.

    6. Example. $N=3, S=6, \text{Die}_1 \leq \text{Die}_2 \leq \text{Die}_3, M=4$


    \[ \boxed{\binom{N+S-M-1}{S-M}} \]
  6. How many combinations are there for \(N\) dice with \(S\) sides \(1, \ldots, S\) that use \(K\) distinct face values? Hint: Let \(X_1, \ldots, X_K\) represent the number of dice with each subscripted face value.

    7. Example. $N=3, S=6, \text{Die}_1 \leq \text{Die}_2 \leq \text{Die}_3, K=1,2,3$


    \[ \boxed{\binom{S}{K}\binom{N-1}{K-1}} \]

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