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Alex wanted to weigh 378 kg of wheat. The weights are available in denominations of 1 kg, 2 kg , 4 kg, 8 kg, 16 kg , 32 kg... etc. He decides not to use more than one weight of each denomination.

In how many ways can he weigh 378 kg, if both sides of the balance can be used for weighing

My attempt :-

Of the 2 sides of balance, i have assumed that the item is kept on right side , now for 1 kg item , i can put 1 kg weight on left side of balance, for 2 kg i can again use left side only, for 3 kg measurement i can use put 1 and 2 kg but how shall I make use of both sides of pan and measure weightsbv how many ways can he weigh 378 kg, if both sides of the balance can be used for weighing

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1 Answer 1

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You have $378=256+64+32+16+8+2$. So, you can put the wheat on one side of the balance and weights of $256$ kg, $64$ kg, $32$ kg, $16$ kg, $8$ kg, and $2$ kg on the other one. And that's the only answer. Because if there was another answer, then you would hav to put the wheat and also some weights on one side of the balance and then the same weights as before ($256$ kg, $64$ kg, $32$ kg, $16$ kg, $8$ kg, and $2$ kg) on the other side, plus the same weights that you have added to the wheat (since every integer can be written as a sum of powers of $2$ in one and only one way). But you want to use each weight exactly once.

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