i have two questions here, both are same in the nature but the solution of the two are different. both questions are solved by "separator" method and i have issue with the formula.
1) 16 oranges are distributed among 4 children such that each gets at least 3 oranges, the number of ways of distributing them is
a. 30 b. 210 c. 15 d. 35 e. 40
after distributing 12 oranges evenly we are left with 4 oranges. Number of grouping simply becomes number of ways to arrange 4 oranges and 3 separators between themselves = (4 + 3)!/(4!*3!) = 35
2) How many of the four-digit numbers with non-zero digits have the sum of their digits as 12?
165
330
132
440
170
This can be done in (11P3)/3! = (11!)/(8! * 3!) = (11*10*9)/6 = 11*15 = 165 ways (WHY IT IS 11P3 WHY NOT 12P3)
my question is that the first question is using total number of sticks and separator as 4+3!, i understand this, but why the second question is using one less separator?
1) 16 oranges are distributed among 4 children such that each gets at least 3 oranges, the number of ways of distributing them is
a. 30 b. 210 c. 15 d. 35 e. 40
after distributing 12 oranges evenly we are left with 4 oranges. Number of grouping simply becomes number of ways to arrange 4 oranges and 3 separators between themselves = (4 + 3)!/(4!*3!) = 35
2) How many of the four-digit numbers with non-zero digits have the sum of their digits as 12?
165
330
132
440
170
This can be done in (11P3)/3! = (11!)/(8! * 3!) = (11*10*9)/6 = 11*15 = 165 ways (WHY IT IS 11P3 WHY NOT 12P3)
my question is that the first question is using total number of sticks and separator as 4+3!, i understand this, but why the second question is using one less separator?
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