Distinct sets

Number Theory Level 3

Let there be five positive integers a a , b b , c c , d d and e e such that a < b < c < d < e a<b<c<d<e and that a + b + c + d + e = 20 a+b+c+d+e=20 . Find the number of such distinct tuples ( a , b , c , d , e ) (a,b,c,d,e) .


The answer is 7.

Avinash S by Avinash S , 22, India

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2 solutions

Avinash S
Jul 22, 2015

thanks samanvay!!!

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Nice question avinash!! So here's the solution . We have 7 cases which satisfy the above mentioned conditions .Given condition :- a+b+c+d+e = 20 and a<b<c<d<e Putting a=1 and b=2 and c=3( lowest positive integers ) ,we get :-

Case 1) 1<2<3<4<10

Case 2) 1<2<3<5<9

Case 3 ) 1<2<3<6<8

Taking a=1 , b=2 and c=4,

Case 4) 1<2<4<5<8

Case 5) 1<2<4<6<7

Taking a=1 and b=3 ,we get -

Case 6 ) 1<3<4<5<7

Taking a= 2 , we get :-

Case 7) 2<3<4<5<6

By observation we find that at a=3 ,the given conditions are not possible . So , only these 7 arrangements satisfy the given conditions . VENI VIDI VICI! :D

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