How many ways can the letters BBBGG be arranged?
Number of ways to arrange 5 elements = 5!.
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 3! to account for the three identical B's and by 2! to account for the two identical G's:
5!/(3!2!) = 10.
BTGmoderatorDC wrote:How many different positive integers having six digits are there, where exactly on of the digits is a 3, exactly two of the digits are a 4, exactly one of the digits is a 5, and each of the other digits is a 7 or an 8?
A) 360
B) 720
C) 840
D) 1,080
E) 1,440
Case 1: The digits are 4, 4, 3, 5, 7, 7
Since the 6-digit arrangement includes two identical 4's and two identical 7's, the number of possible arrangements = 6!/(2!2!) = 180.
Case 2: The digits are 4, 4, 3, 5, 8, 8
Since the 6-digit arrangement includes two identical 4's and two identical 8's, the number of possible arrangements = 6!/(2!2!) = 180.
Case 3: The digits are 4, 4, 3, 5, 7, 8
Since the 6-digit arrangement includes two identical 4's, the number of possible arrangements = 6!/2! = 360.
Total integers = Case 1 + Case 2 + Case 3 = 180+180+360 = 720.
The correct answer is
B.
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