Big answer
Find the number of partitions of 40 if order of integers does matter.
A partition of a positive integer is an expression of as the sum of one or more positive integers (or parts ). The order of the integers in the sum "does not matter": that is, two expressions that contain the same integers in a different order are considered to be the same partition.
For example, 4 has eight partitions: 4 = 3+1 = 1+3 = 2+1+1 = 1+2+1 = 1+1+2 = 1+1+1+1 = 2+2.
The answer is 549755813888.
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1 solution
Let's check partition for any natural number 'n' . Now,
n = 1 + ( n − 1 ) Keeping the last term fixed gives us that only one type of such number is possible.
n = 2 + ( n − 2 ) now, since n-2 is fixed therefore the possible numbers will be the number of partitions of 2 i.e 2.
Similarly, further we can say that the number of partitions of n will be number of partitions of [ ( n − 1 ) + ( n − 2 ) . . . . . . . . + 2 + 1 + 0 ]
Let the number of partitions of n be a(n) a ( n ) = a ( n − 1 ) + a ( n − 2 ) + . . . . . a ( 3 ) + a ( 2 ) + a ( 1 ) + a ( 0 )
now the required series becomes -- 1,1,2,4,8,16,..... now a ( 4 0 ) = 2 3 9 = 5 4 9 7 5 5 8 1 3 8 8 8