Max@Math Revolution wrote:How many arrangements of the letters A, C, C, E, N, T include at least one letter between the two C's?
A. 60
B. 120
C. 240
D. 360
E. 720
Good arrangements = total arrangements - bad arrangements.
Total arrangements:
Number of ways to arrange 6 elements = 6!.
But when an arrangement includes IDENTICAL elements, we must divide by the number of ways each set of identical elements can be ARRANGED.
The reason:
When the identical elements swap positions, the arrangement doesn't change.
Here, we must divide by 2! to account for the two identical C's:
6!/2! = 360.
Bad arrangements:
In a bad arrangement, the two C's are in adjacent slots.
Let [CC] represent the 2 adjacent C's.
Number of ways to arrange the 5 elements [CC], A, E, N and T = 5! = 120.
Good arrangements:
Total arrangements - bad arrangements = 360-120 = 240.
The correct answer is
C.
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