You have one thousand $1 bills. How can you distribute them among ten envelopes

Question

You have one thousand $1 bills. How can you distribute them among ten envelopes so that any amount between $1 and $1000, inclusive, can be given as some combination of these envelopes? No change is allowed, and you are not allowed to open any of the envelopes once you’ve determined how many bills to put in each at the start.

a. For each of the ten envelopes, say specifically how many $1 bills you will put in it. Justify your approach.

b. Explain how you will use (some of) the envelopes to give $142.

c. Explain how you will use (some of) the envelopes to give $917.

Explanation / Answer

given that we have $1 bills and exactly 10 envelopes let us distribute the money in envelopes in 2x form

for x=0

the amount of money will be 20=1

for x=1,2,3,4,5,6,7,8 amount will be $2,$4,$8,$16$32,$64,$128,$256

for x=9 2^9=$512

if add we get (1+2+4+8+16+32+64+128+256+512)=we get $1023

but we $1000 so difference is 1023-1000=$23

so in last enevlope instead of 512 if we place 512-23 =$489 we can distribute 1000

a. For each of the ten envelopes,

we keep $,1$2,$4,$8,$16$32,$64,$128,$256,$489

($1+$2+$4+$8+$16+$32+$64+$128+$256+$489)=$1000

b) for enevoleps to give $142

use $128 enevlope and $8 $4 $2

$142=$8+$2+$4+$128

c) for $917 use enevelope 489,256,128,32,8,4

489+256+128+32+8+4=$917

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