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Question
Five distinct letters are suppose to be transmitted through a communication channel. A total number of 15 blanks is supposed to be inserted between the first and last letter with at least three between every two. The number of ways in which this can be done is
Five distinct letters are suppose to be transmitted through a communication channel. A total number of 15 blanks is supposed to be inserted between the first and last letter with at least three between every two. The number of ways in which this can be done is
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Solution
The correct option is B 2400
5 letters can be arranged among themselves in 5! ways.
Now total 15 blanks are to be inserted between the letters with at least three between every two.
There are 4 spaces between 5 letters. So if we place 3 blanks first between every two, 3 blanks will be left. These three can be placed in any of the 4 spaces between any two letters.
This is equivalent to number of non-negative integral solutions of the equation x1+x2+x3+x4=3.
Number of solutions of above equation is 3+4−1C4−1=6C3=20
Hence, the number of ways in which the task can be completed, using multiplication principle is 5!×20=2400
5 letters can be arranged among themselves in 5! ways.
Now total 15 blanks are to be inserted between the letters with at least three between every two.
There are 4 spaces between 5 letters. So if we place 3 blanks first between every two, 3 blanks will be left. These three can be placed in any of the 4 spaces between any two letters.
This is equivalent to number of non-negative integral solutions of the equation x1+x2+x3+x4=3.
Number of solutions of above equation is 3+4−1C4−1=6C3=20
Hence, the number of ways in which the task can be completed, using multiplication principle is 5!×20=2400
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