Choosing 3 different integers from 1 to 100

Probability Level 4

How many sets of integers are there, such that $1 \leq a < b < c \leq 100$ and $a + b + c$ is a multiple of 3?

The answer is 53922.

2 solutions

Shrihari B
Dec 20, 2014

We can make 3 different sets Set 1 : { 1,4,7,...,100} = 1(mod 3)..... contains 34 elements Set 2 : { 2,5,8,...,98} =2(mod 3)...... contains 33 elements Set 3: {3,6,9,12,...,99} =0(mod 3)...... contains 33 elements

We can choose either all three elements from set 1, 2, 3 or one element from each set. So ways =34C3 + 2 * 33C3 + 34 33 33 =53922

Bill Bell
Aug 25, 2015

Although inelegant as a mathematical solution this is the neatest way I've written code suitable for obtaining explicit solutions to Diophantine-type equations so far. Too bad Python doesn't have macros!

def Triple():
    a=1
    while True:
        b=a+1
        while True:
            c=b+1
            while True:
                yield (a,b,c)
                c+=1
                if c>100:
                    break
            b+=1
            if b>99:
                break
        a+=1
        if a>98:
            break

count=0
for t in Triple():
    count+=1 if sum(t)%3==0 else 0
print count

Choosing 3 different integers from 1 to 100

The answer is 53922.

2 solutions

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