Combinatorics Question.
Last edited by renanmenezes on Sat Feb 27, 2016 6:52 am, edited 1 time in total.
The prompt is asking how many different SUMS OF MONEY between 1¢ and 70¢, inclusive, can be formed using up to ten 2¢ coins and ten 5¢ coins.Matt is touring a nation in which coins are issued in two amounts, 2¢ and 5¢, which are made of iron and copper, respectively. If Matt has ten iron coins and ten copper coins, how many different sums from 1¢ to 70¢ can he make with a combination of his coins?
A. 66
B. 67
C. 68
D. 69
E. 70
Could anyone please take a look at this and explain what the q means by "how many different sums from 1¢ to 70¢ can he make with a combination of his coins?"..
For example, it is possible to form a sum of 16¢ by combining three 2¢ coins and two 5¢ coins:
(3*2) + (2*5) = 16¢.
From 1 to 70, there are 70 possible sums.
The smallest answer choice is 66.
Implication:
At most, 4 of the 70 possible sums cannot be made from the given coins.
Impossible sums are likely to be VERY SMALL or VERY LARGE.
Small sums:
Of the sums between 1 and 10, the following cannot be formed from 2¢ and 5¢ coins:
1 and 3.
Large sums:
If all 20 coins are used, the sum = (10*2) + (10*5) = 70.
The next smallest possible sums are 70-2 = 68 and 70-2-2 = 66.
Thus, 67 and 69 are not achievable.
Since 4 of 70 possible sums -- 1, 3, 67, and 69 -- are not achievable, the total number of sums that can be formed = 70-4 = 66.
The correct answer is A.
