As usual, you need to provide explanations, not just answers. B15 Find the numbe

Question

As usual, you need to provide explanations, not just answers.

B15 Find the number of ways to rearrange three As and seven Bs if no two As can appear consecutively. (Hint: Start with BBBBBBB and consider where the three As can be inserted. There are eight places.) 18 COMBINATORICS B16* Use the method suggested by problem B15 to fill in the answer to the following standard problem Standard Problem #4 Find the number of bit strings that contain a given number of Os and Is, such that there are no two consecutive s Let m be the number of Os and n be the number of 1s. What is the answer, in of m and n? B17 Find the number of ways to select three distinct digits from the set 0, 1,2,... ,9 if no two consecutive digits can be selected. (Hint: This has something to do with problem B15.)

Explanation / Answer

B15:

There are 8 places in BBBBBBB where we can place A. Six palces are between B's and two spaces one at starting of string and second in last of string.

First A can be placed in 8 ways, second in 7 ways and third in 6 ways. So possible number of ways rearrging three A's and 7 B's is

8*7*6 = 336

B16:

Since theer are m 0's so number of places for 1's is (m+1). So first A can be placed is (m+1) ways, second in m places ,...last in (m-n+2) ways.

So possible number of ways of arranging string is

(m+1) * m * (m-1) *... * (m-n+2)

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