Question

A bank allows one person to have only one savings account insured to $100,000.However, a larger family may have accounts for each individual, and also accountingthe names of any 2 people, any 3 and so on. How many accounts are possible for afamily of 2? Of 3? Of 5? Of? Hint: See Problem 2.

Short answer

Answer

The number of accounts for a family of 2 is 3,family of 3 is 7,family of 5 is 31 and a family of n is $2^n-1$.

Step-by-step solution

Given Information

The total number of combinations ofthings taken 1, 2, 3, · · · ,at a time, is$2^n-1$.

Definition of Independent Event

When the order of arrangement is definite, the permutation is applied and when the order is not definite, combination is applied.

Find the total number of accounts for family of 2,3, 5 and 

Substitute 2 for n into$2^n-1$and simplify to obtain the number of accounts for a family of 2.

$$\begin{gathered} 2^2-1-4-1 \\ -3 \end{gathered}$$

Substitute 3 for n into$2^n-1$and simplify to obtain the number of accounts for a family of 3.

$$\begin{gathered} 2^3-1=8-1 \\ -7 \end{gathered}$$

Substitute 5 for n into$2^n-1$and simplify to obtain the number of accounts for a family of 5.

$$\begin{gathered} 2^5-1=32-1 \\ =31 \end{gathered}$$

It can be observed that the number of accounts for a family of n is $2^n-1$.