DISCRETE MATH Answer Exercise 1.32 where 12 symbols being transmitted are four A

Question

DISCRETE MATH

Answer Exercise 1.32 where 12 symbols being transmitted are four A's, four B's, and four C's.

3) ,4,43) bb 4) 4,4,4 4) lbbb 3) 2.2, 2. 3. 3. 3 4) 4, 4, 4. 4, 4,4 4) 1loooo o o 4) 4,4,4,4,4,44) 0o0 A message is made up of 12 different symbols and is to be transmitted through a com- munication channel. In addition to the 12 symbols, the transmitter will also send a total of 45 (blank) spaces between the symbols, with at least three spaces between each pair of consecutive symbols. In how many ways can the transmitter send such a message? There are 12! ways to arrange the 12 different symbols, and for each of these arrangements there are 11l positions between the 12 symbols. Because there must be at least three spaces between successive symbols, we use up 33 of the 45 spaces and must now locate the remaining 12 spaces. This is now a selection, with repetition, of size 12 (the spaces) from a collection of size 1 1 (the locations), and this can be accomplished in C( 1 1 + 12-1, 12) = 646,646 ways.

Explanation / Answer

There are total of 33 white spaces as told by the person and remaining 12 places from a collection of size 11, which can be done in 22C12 ways as shown above

Now we have 12 empty locations for 4A's,4B's and 4C's

We can choose four position for A's with choosing probability 12C4

Now there are 8 empty locations for 4B's and 4C's

We can choose four position for B's with choosing probability 8C4

Now there are 4 empty locations for 4C's

We can choose four position for C's with choosing probability 4C4

Final Answer = 22C12 * 12C4 * 8C4 * 4C4

=> 22406283900 ways

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