Using the following bills: $1, $5, $10, $20, $50. How many unique sums

Question

Using the following bills: $1, $5, $10, $20, $50. How many unique sums can be formed using any number of these bills only once?

(A) 20

(B) 24

(C) 31

(D) 75

(E) 120

chetan2u · Accepted Answer

Short answer

Each of the bill/note can have two options: used or not used, so 2 ways. 
For all five of them: 2*2*2*2*2 = 32
But this contains case when none is picked so subtract that 32-1=31

Combinations

1) Only 1 bill used: 5 ways
2) Any two bills used: 5C2 or 10 ways
3) Any three bills used: 5C3 or 10 ways
4) Any four bills used: 5C4 or 5 ways
5) All five used: 5C5 or 1 way
Total: 5+10+10+5+1 or 31 ways

C

Abhishekgmat87 · Answer

If we take any number of the given bills, the sum will be unique.

We can take any 1 bill in 5C1 ways or 5 ways
2 bills can be taken in 5C2 or 10 ways
3 bills can be taken in 5C3 or 10 ways
4 bills can be taken in 5C4 or 5 ways
5 bills can be taken in 5C5 or 1 way

Total ways - 5 + 10 + 10 + 5 + 1 = 31

Option C

sanjitscorps18 · Answer

The word "unique sums" is tricky here. Two meanings can be taken out 
1. sum of money
2. sums of values

Thankfully if we consider the sums of values, the corresponding answer is not part of the choices.

So we have
Single values = 5
Taking 2 at a time = 5C2 = 10
Taking 3 at a time = 5C3 = 10
Taking 4 at a time = 5C4 = 5
Taking 5 at a time = 5C5 = 1

Overall we have 31 Sums

Option C

botuser · Answer

Each order is not unique so use Combinations and we can select each item at a time or all items in a single instance or anything in between.

5C1= 5
5C2= 10
5C3= 10
5C4= 5
5C5= 1

Total 31, So C

luisdicampo · Answer

Deconstructing the Question  
We are given the bills:
 1, 5, 10, 20, 50 .
Each bill can be used at most once.
We are asked how many unique sums can be formed.

Step-by-step  
Each of the 5 bills can either be included or excluded.
Total subsets:
 2^5 = 32

Exclude the empty subset (sum = 0):
 32 - 1 = 31

Each bill is greater than the sum of all smaller bills:
 5 > 1 
 10 > 1+5 
 20 > 1+5+10 
 50 > 1+5+10+20

Thus every subset produces a unique sum.

Answer: 31 (C)

Using the following bills: $1, $5, $10, $20, $50. How many unique sums

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GMAT Problem Solving (PS) Questions

Using the following bills: $1, $5, $10, $20, $50. How many unique sums

Prep Toolkit

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