Step-by-step explanation: To distribute 100 $1 bills among 7 envelopes so that the emcee can give away any amount from $1 to $100 without redistributing the bills, the emcee can follow the following steps:
1. First, the emcee needs to assign each envelope a unique power of 2 starting from 1 (1, 2, 4, 8, 16, 32, 64). These powers of 2 will represent the possible amounts that can be given using combinations of envelopes.
2. Then, the emcee can distribute the 100 $1 bills by placing a specific number of bills in each envelope corresponding to the powers of 2. The envelopes would contain the following number of bills:
- Envelope 1: 1 bill
- Envelope 2: 2 bills
- Envelope 3: 4 bills
- Envelope 4: 8 bills
- Envelope 5: 16 bills
- Envelope 6: 32 bills
- Envelope 7: 37 bills
3. By having the bills distributed in this way, the emcee can cover all amounts from $1 to $100 by simply giving the contestant the envelopes corresponding to the amounts needed. For instance, to give $68, the emcee would hand over envelopes 4, 6, and 7 (representing 8, 32, and 37 bills respectively), totaling $68.
4. This method ensures that any amount from $1 to $100 can be formed without the need to rearrange the bills, providing a flexible and efficient way to give away the cash prizes at the carnival.
By organizing the bills in this manner, the emcee can guarantee that any amount from $1 to $100 can be created by choosing the right combination of envelopes, simplifying the process of giving away the cash prizes.
