a b c d e = 1 0 5 0
and a, b, c, d, e are positive integers. How many ordered tuples ( a , b , c , d , e ) exist?
Extra Credit: How many unordered sets a , b , c , d , e exist?
Extra Credit: How many ordered tuples ( a , b , c , d , e ) exist such that a≠b≠c≠d≠e?
Unrelated Credit: Solve the Aunty's Teacups problem
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The easier way to express your ideas, is that we want to find non-negative integer solutions to:
a 2 + b 2 + c 2 + d 2 + d 2 a 3 + b 3 + c 3 + d 3 + d 3 a 5 + b 5 + c 5 + d 5 + d 5 a 7 + b 7 + c 7 + d 7 + d 7 = 1 = 1 = 2 = 1
where a n denotes the prime power exponent of n in a .
Hence, there are ( 1 5 ) 3 × ( 2 6 ) ways.
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Exactly How I did it.
Me too Exactly same.
what if we were asked to find number of unordered tuples???
I Reason it in this way :
a
b
c
d
e
=
2
×
3
×
5
2
×
7
Now This situation is analogous to, There are
: 1 book type-1 (say 2)
: 1 book type-2 (say 3)
: 2 book type-3 (say 5)
: 1 book type-4 (say 7)
"Here Book of Same type are identical "
and there are 5 beggar's , and we have to distribute these books to them .
Book-1 can be distributed to 5 beggers in ( 5 4 ) ways !
Book-2 . . . . . . . . . . . . . . ( 5 4 ) ways !
Book-3 ............................ ( 6 4 ) ways!
Book-4........................... ( 5 4 ) ways !
So Total no of ways = ( 5 4 ) 3 × ( 6 4 )
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You have a box [2,3,5,5,7]
You have initialised a=b=c=d=e=1
You have to take away a number from the box and multiply it with one of a, b, c or d and repeat until the box is exhausted.
In how many ways can you do this?
The numbers 2,3,7 can be multiplied with any of a,b,c,d,e. So there are 5^3 ways of doing it.
Either you multiply 5^2 to one of the numbers which can be done in 5 ways or you throw the two 5's at two two different variables which can be done in 10 ways.
So you can do it in a total of 5^3(10+5) ways.