Finding The Total Number of Printer Combinations

Abstract

We are presented with a problem of a “printer” which takes a combination of short and long “pulses”, similar to that of Morse code, to output a desired sheet of material. We are given 10 “spaces” to enter our code, however the long pulse takes up 2 spaces on this theoretical space budget.

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Problem Statement

Consider a sequence formed by two types of symbols:

• A short symbol $S$, which takes 1 unit of space.
• A long symbol $L$, which takes 2 units of space.

The total available space for the sequence is at most 10 units. Determine the total number of distinct sequences of $S$ and $L$ that can be formed, accounting for sequences of lengths ranging from 1 to 10 units under the following constraints:

1. A sequence may include at most $k$ long symbols, where $2k\le n$, and $n$ is the sequence’s total length.
2. The remaining space after placing $k$ long symbols is filled with short symbols.

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Solution

Step 1: General Setup

For a sequence of total length $n$, where $1\le n\le 10$:

• Each long symbol $L$ takes 2 units of space. If $k$ is the number of long symbols, they consume $2k$ units of space.
• The remaining space, $n-2k$, is filled with short symbols $S$. Since each short symbol takes 1 unit of space, the number of short symbols is $n-2k$.
• The total number of symbols in the sequence is $k+(n-2k)=n-k$.

The number of ways to arrange $k$ long symbols $L$ and $n-2k$ short symbols $S$ is given by the binomial coefficient:

$$\left(\genfrac{}{}{0ex}{}{n-k}{k}\right).$$

Step 2: Constraints on $k$ for a Fixed $n$

For a fixed sequence length $n$, the number of long symbols $k$ must satisfy:

$$2k\le n\text{or equivalently}k\le \lfloor n/2\rfloor .$$

Thus, for each $n$, $k$ ranges from 0 to $\lfloor n/2\rfloor$.

The total number of sequences for a fixed $n$ is:

$$\sum _{k=0}^{\lfloor n/2\rfloor }\left(\genfrac{}{}{0ex}{}{n-k}{k}\right).$$

Step 3: Summing Over All Possible $n$

To account for all possible sequence lengths, we sum over all $n$ from 1 to 10:

$$\text{Total combinations}=\sum _{n=1}^{10}\sum _{k=0}^{\lfloor n/2\rfloor }\left(\genfrac{}{}{0ex}{}{n-k}{k}\right).$$

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Step-by-Step Calculation

We compute the total number of sequences for each $n$ and $k$:

For $n=1$:

• $k=0$: Only 1 short symbol.

  $$\left(\genfrac{}{}{0ex}{}{1}{0}\right)=1$$

Total for $n=1$: $1$

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For $n=2$:

• $k=0$: 2 short symbols.

  $$\left(\genfrac{}{}{0ex}{}{2}{0}\right)=1$$

• $k=1$: 1 long symbol.

  $$\left(\genfrac{}{}{0ex}{}{1}{1}\right)=1$$

Total for $n=2$: $1+1=2$

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For $n=3$:

• $k=0$: 3 short symbols.

  $$\left(\genfrac{}{}{0ex}{}{3}{0}\right)=1$$

• $k=1$: 1 long symbol and 1 short symbol.

  $$\left(\genfrac{}{}{0ex}{}{2}{1}\right)=2$$

Total for $n=3$: $1+2=3$

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For $n=4$:

• $k=0$: 4 short symbols.

  $$\left(\genfrac{}{}{0ex}{}{4}{0}\right)=1$$

• $k=1$: 1 long symbol and 2 short symbols.

  $$\left(\genfrac{}{}{0ex}{}{3}{1}\right)=3$$

• $k=2$: 2 long symbols.

  $$\left(\genfrac{}{}{0ex}{}{2}{2}\right)=1$$

Total for $n=4$: $1+3+1=5$

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For $n=5$:

• $k=0$: 5 short symbols.

  $$\left(\genfrac{}{}{0ex}{}{5}{0}\right)=1$$

• $k=1$: 1 long symbol and 3 short symbols.

  $$\left(\genfrac{}{}{0ex}{}{4}{1}\right)=4$$

• $k=2$: 2 long symbols and 1 short symbol.

  $$\left(\genfrac{}{}{0ex}{}{3}{2}\right)=3$$

Total for $n=5$: $1+4+3=8$

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For $n=6$:

• $k=0$: 6 short symbols.

  $$\left(\genfrac{}{}{0ex}{}{6}{0}\right)=1$$

• $k=1$: 1 long symbol and 4 short symbols.

  $$\left(\genfrac{}{}{0ex}{}{5}{1}\right)=5$$

• $k=2$: 2 long symbols and 2 short symbols.

  $$\left(\genfrac{}{}{0ex}{}{4}{2}\right)=6$$

• $k=3$: 3 long symbols.

  $$\left(\genfrac{}{}{0ex}{}{3}{3}\right)=1$$

Total for $n=6$: $1+5+6+1=13$

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For $n=7$:

• $k=0$: 7 short symbols.

  $$\left(\genfrac{}{}{0ex}{}{7}{0}\right)=1$$

• $k=1$: 1 long symbol and 5 short symbols.

  $$\left(\genfrac{}{}{0ex}{}{6}{1}\right)=6$$

• $k=2$: 2 long symbols and 3 short symbols.

  $$\left(\genfrac{}{}{0ex}{}{5}{2}\right)=10$$

• $k=3$: 3 long symbols and 1 short symbol.

  $$\left(\genfrac{}{}{0ex}{}{4}{3}\right)=4$$

Total for $n=7$: $1+6+10+4=21$

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For $n=8$:

• $k=0$: 8 short symbols.

  $$\left(\genfrac{}{}{0ex}{}{8}{0}\right)=1$$

• $k=1$: 1 long symbol and 6 short symbols.

  $$\left(\genfrac{}{}{0ex}{}{7}{1}\right)=7$$

• $k=2$: 2 long symbols and 4 short symbols.

  $$\left(\genfrac{}{}{0ex}{}{6}{2}\right)=15$$

• $k=3$: 3 long symbols and 2 short symbols.

  $$\left(\genfrac{}{}{0ex}{}{5}{3}\right)=10$$

• $k=4$: 4 long symbols.

  $$\left(\genfrac{}{}{0ex}{}{4}{4}\right)=1$$

Total for $n=8$: $1+7+15+10+1=34$

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For $n=9$:

• $k=0$: 9 short symbols.

  $$\left(\genfrac{}{}{0ex}{}{9}{0}\right)=1$$

• $k=1$: 1 long symbol and 7 short symbols.

  $$\left(\genfrac{}{}{0ex}{}{8}{1}\right)=8$$

• $k=2$: 2 long symbols and 5 short symbols.

  $$\left(\genfrac{}{}{0ex}{}{7}{2}\right)=21$$

• $k=3$: 3 long symbols and 3 short symbols.

  $$\left(\genfrac{}{}{0ex}{}{6}{3}\right)=20$$

• $k=4$: 4 long symbols and 1 short symbol.

  $$\left(\genfrac{}{}{0ex}{}{5}{4}\right)=5$$

Total for $n=9$: $1+8+21+20+5=55$

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For $n=10$:

• $k=0$: 10 short symbols.

  $$\left(\genfrac{}{}{0ex}{}{10}{0}\right)=1$$

• $k=1$: 1 long symbol and 8 short symbols.

  $$\left(\genfrac{}{}{0ex}{}{9}{1}\right)=9$$

• $k=2$: 2 long symbols and 6 short symbols.

  $$\left(\genfrac{}{}{0ex}{}{8}{2}\right)=28$$

• $k=3$: 3 long symbols and 4 short symbols.

  $$\left(\genfrac{}{}{0ex}{}{7}{3}\right)=35$$

• $k=4$: 4 long symbols and 2 short symbols.

  $$\left(\genfrac{}{}{0ex}{}{6}{4}\right)=15$$

• $k=5$: 5 long symbols.

  $$\left(\genfrac{}{}{0ex}{}{5}{5}\right)=1$$

Total for $n=10$: $1+9+28+35+15+1=89$

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Final Total

Summing all sequences:

$$1+2+3+5+8+13+21+34+55+89=231$$

Final Answer: 231 distinct sequences.