题目信息
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?
A:96
B:120
C:144
D:180
E:240
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已选答案：
正确答案：
E:240
Arithmetic Elementary combinatorics
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, m = 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
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).
Tip The alternative approach for finding m is useful when a direct count of the number of suitable ways is more difficult than a direct count of the number of non-suitable ways. An example is determining the number of 8-letter strings that can be formed from the letters in REPEATED in which there is at least one pair of E's having at least one other letter between them. For this example, m =
= 56 − 6 = 50.
The value of n is equal to the number of ways to place the 4 remaining letters into 4 positions, where order matters and the letters are selected without replacement. Thus, n = 4! = 24.
Therefore, the answer is m × n = 10 × 24 = 240.
The correct answer is E.
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:
| C*C*** | C**C** | C***C* | C****C |
There are 3 more possibilities when a C is in the second position:
| *C*C** | *C**C* | *C***C |
There are 2 more possibilities when a C is in the third position:
| **C*C* | **C**C |
There is 1 more possibility when a C is in the fourth position:
| ***C*C |
Therefore, m = 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
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).| CC**** | *CC*** | **CC** | ***CC* | ****CC |
Tip The alternative approach for finding m is useful when a direct count of the number of suitable ways is more difficult than a direct count of the number of non-suitable ways. An example is determining the number of 8-letter strings that can be formed from the letters in REPEATED in which there is at least one pair of E's having at least one other letter between them. For this example, m =
= 56 − 6 = 50.The value of n is equal to the number of ways to place the 4 remaining letters into 4 positions, where order matters and the letters are selected without replacement. Thus, n = 4! = 24.
Therefore, the answer is m × n = 10 × 24 = 240.
The correct answer is E.
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