# A person has four notes of rupees whose denominations are 1,2,5,10. Find the number of different sums of money she can form from them.

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### 1 Answer

We have 4 different notes: 1,2,5,10 , and we are asked to find the number of different sums we can form from them.

(1) If we use only one note there are 4 sums: 1,2,5, and 10.

(2) If we use 2 notes: We could list all possibilities,i.e. (1,2),(1,5),(1,10),(2,5),(2,10),(5,10) for 6 different sums: 3,6,11,7,12,15.

We could also note that for the first note chosen we have 4 choices, and for the second note chosen we have 3 choices; from the fundamental counting principle we have 12 choices.However, the choice of 1 then 5 is the same as the choice of 5 then 1: each pair has a duplicate, so the total number of choices is 12/2=6. This is the same as `_4C_2=6` (or C(4,2)=6) where C denotes a combination. We don't use permutations as order doesn't matter.

(3) Taking 3 notes: `_4C_3=4` so there are 4 ways to choose 3 notes: (1,2,5),(1,2,10),(1,5,10),(2,5,10) which give the sums: 8,13,16,17

(4) Taking all 4 notes: the sum is 18.

**(5) If we take no notes the sum is 0.

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**There are 16 different sums possible:**

**0,1,2,3,5,6,7,8,10,11,12,13,15,16,17,18 rupees.**

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`_4C_0+_4C_1+_4C_2+_4C_3+_4C_4=1+4+6+4+1=16`

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