combination problem

[absvalue]

This puzzle has me scratching my head:

A special type of door lock has a panel with five buttons labeled with the digits 1 through 5. This lock is opened by a sequence of three actions. Each action consists of either pressing one of the buttons or pressing a pair of them simultaneously. For example, 12-4-3 is a possible combination. The combination 12-4-3 is the same as 21-4-3 because both the 12 and the 21 simply mean to press buttons 1 and 2 simultaneously.

a. How many combinations are possible?
b. Hom many combinations are possible if no digit is repeated in the combination?

I started out by isolating the 'single digit' and 'multiple digit' combinations. If you entertain only single digit actions, there are 5^3 possibilities. Likewise, if you entertain only multiple digit actios, there are 5!^3 possibilities. So, there are 5^3*5!^3 total possibilities for part a. Is this correct?

I'm not sure where to start with part b. Any hints would be greatly appreciated.

 

[soroban]

Hello, absvalue!

I think I have part (a) . . .

A special type of door lock has a panel with five buttons labeled with the digits 1 through 5.
This lock is opened by a sequence of three actions.
Each action consists of either pressing one of the buttons or pressing a pair of them simultaneously.
For example, 12-4-3 is a possible combination.
The combination 12-4-3 is the same as 21-4-3, because both the 12 and the 21 mean "press 1 and 2 simultaneously".

a. How many combinations are possible?

Your approach is correct!

Let \(\displaystyle S\) = single-digit, .\(\displaystyle D\) = double-digit.

\(\displaystyle \text{There are: }\:5\text{ choices for an }S.\)

\(\displaystyle \text{There are: }\:{5\choose2} = 10\text{ choices for a }D.\)

\(\displaystyle \text{There are four possible cases: }\;\begin{Bmatrix}\text{3 S's} \\ \text{3 D's} \\ \text{1 S, 2 D's} \\ \text{2 S's, 1 D} \end{Bmatrix}\)

\(\displaystyle \text{3 S's}\)
\(\displaystyle \text{There are: }\;5^3 \:=\:125 \text{ combinations.}\)


\(\displaystyle \text{3 D's}\)
\(\displaystyle \text{There are: }\:10^3 \:=\:1000\text{ combinations.}\)


\(\displaystyle \text{1 S, 2 D's}\)
\(\displaystyle \text{These can occur in 3 ways: }\:SDD,\SD,\DS\)
. . \(\displaystyle \text{For example, }SDD\text{ has: }\:5\cdot10^2 \:=\:500\text{ ways.}\)
\(\displaystyle \text{Hence, } \:\text{(1 S, 2 D's) has: }\:3\cdot500 \:=\:1500\text{ combinations.}\)


\(\displaystyle \text{2 S's, 1 D}\)
\(\displaystyle \text{There can occur in 3 ways: }\:SSD,\:SDS,\SS\)
. . \(\displaystyle \text{For example, }SSD\text{ has: }\:5^2\cdot10 \:=\:250\text{ combinations.}\)
\(\displaystyle \text{Hence, }\:\text{(2 S's, 1 D) has: }\:3\cdot250 \:=\:750\text{ combinations.}\)


\(\displaystyle \text{Therefore, there are: }\;125 + 1000 + 1500 + 750 \;=\;3375\text{ combinations.}\)