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Mathematics-Online problems: | ||

## Solution to the problem of the (previous) week |

** Problem:**

In a PIN consisting of 5 characters, at least two special symbols (, , ) are used apart from the digits 0 to 9 due to security reasons.

**a)** How many such PINs do exist?

How many of these have

**b)** only different special characters, e. g. 0
9
,

**c)** at least two identical special characters, e. g.
5
,

**d)** at least two identical digits, e. g. 33
3 ?

**Answer:**

a) |
||

b) |
||

c) |
||

d) |

** Solution:**

2 SC: | = | ||

3 SC: | = | ||

4 SC: | = | ||

5 SC: | = | ||

So, the total number is |

**b)**
If only different special characters are allowed, we get a modified table:

2 SC: | = | ||

3 SC: | = | ||

i. e., there exist of these PINs. |

**c)**
From the total number of PINs, we have to subtract those with only different special characters. Hence, there are

of such PINs.

This result can also be obtained, if in each case of table a) we subtract the number of PINs with only different special characters:

2 SC: | = | ||

3 SC: | = | ||

4 SC: | = | ||

5 SC: | = | ||

So, these numbers amount to |

**d)**
Analogous to step **c)** we obtain

2 SC: | = | ||

3 SC: | = | ||

the sum being |

[problem of the week]