If abc is a three digit number and a, b, c are distinct digits, what is abc if acb+bca+bac+cab+cba=3194

we have (In digit form)
abc = 100a + 10b + c
acb = 100 a + 10c + b
bac = 100b + 10a + c
bca = 100b + 10c + a
cab = 100c + 10 a + b
cba = 100c + 10 b + a

so sum = 222 ( a+ b+ c) > 3194 and 3194 + abc = 222(a+b+c)


find the lower and upper limit of a+b+ c

 a + b + c > 3194/222 = 14.38
so lower limit = 15 and let us start and lowe limit and go one by one finding abc and checking

if a + b+ c = 15 then abc = 222*15-3194 = 136 does not satisfy
a + b + c = 16 , abc = 106 + 222 = 358 yes
a+ b+ c = 17, abc = 358+ 222 = 780 no
a+ b+ c = 18, abc= 780+ 222 > 1000 = 772 no and a+b+ c > 18 is not possible

so abc = 358