In how many ways can we position 6 into oddered summands ? [for exp,3 can be positioned into 3 ways as 1+2, 2+1 and 1+1+1] :
1)27
2)29
3)31
4)33
(how to do this math????)

Dear Student,
Please find below the solution to the asked query:

2 summands to get a 6 can be of 5 types: 1,5,2,4,3,3,4,2,5,13 summands to geta 6 can be of 3 types:1,1,4 or 2,2,2 or 1,2,3Now, 1,1,4 can be arranged P23 ways=3!3-2!=3Similarly, 1,2,3 can be arranged P33 ways=3!3-3!=6Thus, Total 6+3+1=10 ordered summands when 6 is obtained by 3 summands.4 summands to geta 6 can be of 2 types:1,1,1,3, 1,1,2,2Now, 1,1,1,3 can be arranged P34 ways=4!4-3!=4Similarly, 1,1,2,2 can be arranged C24 ways=4!2!2!=6Thus, Total 4+6=10 ordered summands when 6 is obtained by4 summands.5 summands to geta 6 can be of 1 types:1,1,1,1,2Now, 1,1,1,1,2 can be arranged P45 ways=5!5-4!=56 summands to geta 6 can be of 1 types:1,1,1,1,1,1Thus, total number of ways=5+10+10+5+1=31
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