Chapter 15: Q3P (page 742)

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.

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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

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

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

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

$2^{2} - 1 - 4 - 1 - 3$

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

$2^{3} - 1 = 8 - 1 - 7$

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

$2^{5} - 1 = 32 - 1 = 31$

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

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