单选题
The letters C, I, R, C, L, and E can be used to form 6-letter strings such as CIRCLE or CCIRLE. Using these letters, how many different 6-letter strings can be formed in which the two occurrences of the letter C are separated by at least one other letter?
【正确答案】
E
【答案解析】This can be solved by using the Multiplication Principle. The answer is m × n, where m is the number of ways to choose the 2 suitable positions in which to place the C’s and n is the number of ways in which to place the 4 remaining letters in the 4 remaining positions.The value of m can be found by a direct count of the number of suitable ways to choose the 2 positions in which to place the C’s. In what follows, each * denotes one of the 4 remaining positions.There are 4 possibilities when a C is in the first position:
There are 3 more possibilities when a C is in the second position:
There are 2 more possibilities when a C is in the third position:
There is 1 more possibility when a C is in the fourth position:
Therefore, w = 4 + 3 + 2 + 1 = 10.Alternatively, the value of m can be found by subtracting the number of non-suitable ways to place the C’s (i.e., the number of consecutive positions in the string) from the number of all possible ways to place the C’s (suitable or not).This gives m = 15 - 5 = 10, where 15 =
is the number of all possible ways to place the C’s ("6 choose 2") and 5 is the number of non-suitable ways to place the C’s (shown below).
There are 3 more possibilities when a C is in the second position:There are 2 more possibilities when a C is in the third position:There is 1 more possibility when a C is in the fourth position:Therefore, w = 4 + 3 + 2 + 1 = 10.Alternatively, the value of m can be found by subtracting the number of non-suitable ways to place the C’s (i.e., the number of consecutive positions in the string) from the number of all possible ways to place the C’s (suitable or not).This gives m = 15 - 5 = 10, where 15 = is the number of all possible ways to place the C’s ("6 choose 2") and 5 is the number of non-suitable ways to place the C’s (shown below).