Question

a delicatessen makes a roast - beef - on - sourdough sandwich for which you can choose from eight condiments.
(a) how many different types of roast - beef - on - sourdough sandwiches can the delicatessen prepare?
(b) what is the minimum number of condiments the delicatessen must have available if it wishes to offer at least 1,000 different types of roast - beef - on - sourdough sandwiches?
a company that builds homes advertises that, by choosing from its upgrade options, each of its new homes is available in 128 different variations. how many upgrade options does the company offer?

Explanation:

Step1: Calculate total sandwich variations

Each condiment has 2 choices: include or exclude. For 8 condiments, the total number of combinations is $2^n$, where $n=8$.
$2^8 = 256$

Step2: Solve for minimum condiment count

We need $2^n \geq 1000$. Test values of $n$:
$2^9=512$, $2^{10}=1024$. Since $1024\geq1000$, $n=10$.

Step3: Find home upgrade options

Total variations = $2^n=128$. Solve for $n$:
$2^n=128$, so $n=\log_2 128=7$.

Answer:

(a) 256
(b) 10

1. 7