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Question:

A message wants to be sent through a transmitter. There are 6 distinct symbols (Assume A B C D E F). There is also 50 blank spaces given, with at-least 8 spaces between each successful consecutive symbol.

How many ways the message be sent ?

My solution:

There are 6 symbols given, and their order clearly matters.

$\therefore$ 6!

Also, A (1) B (2) C (3) D (4) E (5) F, there are 5 locations in between the symbols. Each consecutive pair of symbol can have at-least 8 spaces in between.

50 (total spaces) - (5(locations) * 8(spaces)) = 10 left over blank spaces that can be placed anywhere.

$\therefore$ $n+r-1 \choose r$ using combinations with repetitions.

n = 5 locations and r = 10 leftover spaces (limiting factor)

$\therefore$ $5+10-1 \choose 10$

Putting everything together we get:

$\therefore$ 6! * $5+10-1 \choose 10$ ways to arrange the symbols and leftover blank spaces.

Logical Question:

Am I right with my logic?

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    Feel free to post your solution even if the answers are the same, it is still helpful! and I will definitely up-vote it if it provides confirmation! – 2017-02-23

2 Answers 2

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The way I like to handle enforced gaps is to "glue" $8$ blanks to each of the symbols. Say that the symbols are ordered $ABDEF$ (the $6!$ takes care of the other orderings. I would glue $8$ blanks to each of $A,B,C,D$ and $E$ (but $F$ does not need any blanks since it could even be the last thing transmitted.

So now we have $6$ "super-letters" and $50-5\cdot 8 = 10$ free blanks. That is, there are $16$ such objects and we can put them into $16$ slots, choosing any $6$ slots for our "super-letters." Once we have made this choice, the arrangement is fully determined.

So the answer to the problem will be $$6!\binom{16}{6} = 6! \cdot 8008 = 5765760$$

Where you went wrong is that there are seven, not five, possible locations for a given free blank: Before $A$, between $A$ and $B \ldots$, between $E$ and $F$, and after $F$.

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    Mark, could you please clarify what you mean by 6 "super-letter" ? I have a similar textbook question and solution like mine, can I upload that? I believe it can back up my answer. – 2017-02-23
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Similar Textbook Question and Solution:

  • Reference:

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    @Mark Fischler, please take a look at this example from my textbook. I believe this provides a great clarification. Tell me what you think. – 2017-02-23