In the land of Binary, the unit of currency is called Ben and currency notes are available in denominations 1,2 , 2^2, 2^3, . Bens. The rules of the Government of Binary stipulate that one can not use more than two notes of any one denomination in any transaction. For example, one can give a change for 2 Bens in two ways: 2 one Ben notes or 1 two Ben note. For 5 Ben one can give 1 one Ben note and 1 four Ben note or

Step-by-step solution

1,2,2^2 . a+2 b+4 c+ =100 0 a 2,0 b 2 aligned & a=0 & & b+2 c+ =50 aligned or, aligned & a=1 & 2(b+2 c+ )=99 & a=1 (not possible) aligned aligned & a=2 & & b+2 c+ =49 & c+2 d+ =24 aligned Let number of solutions be a₁₀₀ aligned & a₁₀₀=a₅₀+a₂₄ & & aligned a₁₀₀= & (a₂₅+a₂₄ )+a₂₄=a₂₅+2 a₂₄ & = & a₁₂+2 (a₁₂+a₅ ) aligned aligned aligned & =3 a ₁₂+2 a ₅ & =3 (a₆+a₂ )+2 (a₂ ) & =5 a₂+3 a₆ & =5 a₂+3 (a₃+a₂ ) & =8 a₂+3 a₃ aligned For a₃ Let, a+2 b+4 c=3a=1 2 b+4 c=2 aligned & b+2 c=1 & b=1, c=0 aligned (1,1,0) only one solu