how many compositions does the integer 15 have whose first p

how many compositions does the integer 15 have whose first part is not 1?

Solution

There are 2ⁿ¹ compositions of n 1

there fore 15 has  2¹⁴ compositions

given first part is not 1

=> subtracting partitions with 1 as first part but they are equal to partitions of 14 i.e., 2¹³

there fore

compositions does the integer 15 have whose first part is not 1 are 2¹⁴-2¹³=2¹³(2-1)=2¹³