Of the 5 distinguishable wires that lead into an apartment, 3 are for cable television service, and 3 are for telephone service. Using these wires, how many distinct combinations of 3 wires are there such that at least 1 of the wires is for cable television? Choose the correct choice
A) 6
B) 7
C) 8
D) 9
E) 10
Given:   Total number of wires = 5   Number of cable wires = 2   Number of telephone wires = 3   To find:   Number of combination which has at least one cable wires   Solution:   No of ways of selecting'at least'1 cable wire means, we can select more than one as well. The minimum we can select is one and the maximum we can select, given the constraints that 3 wires need to be selected in total and there are 2 cable wires, is 2   Since it is a combination of wires, the arrangement is not important   Approach 1:   Number of ways of selecting at least one cable wire in a selection of 3 wires from 5 wires =Selection 1(Number of ways of selecting one cable wire and two telephone wires )+Selection 2(Number of ways of selecting two cable wires and 1 telephone wire)   Selection 1   Number of ways of selecting one cable wire = 2C1 = 2   Number of ways of selecting 2 telephone wires = 3C2 = 3   Total = 2C1 * 3C2 = 6 ( m ways of doing something and n ways of doing something else together give m*n ways of doing - the holy grail rule in Combinatorics)   Selection 2   Number of ways of selecting one cable wire = 2C2 = 1   Number of ways of selecting 2 telephone wires = 3C1 = 3   Total = 2C2 * 3C1 = 3 ( m ways of doing something and n ways of doing something else together give m*n ways of doing - the holy grail rule in Combinatorics)   Selection 1 + Selection 2 = 9 ways of selecting 3 wires out of 5 such that at least one is a cable wire   Approach 2   Number of ways of selecting 3 wires out of 5 such that at least one is a cable wire =Selection X(Total number of ways of selecting 3 wires from the 5) -Selection Y(total ways of selecting 3 wires such that none is a cable i.e all the three are telephone wires)   Total number of ways of selecting 3 wires out of 5 = 5C2 = 10   Number ways of selecting 3 wires such that none is a cable i.e all the three are telephone wires = 3C3 ( 3 telephone wires and we are selecting all the three at once) = 1   Selection X - Selection Y = 7   Answer is Option 7
<result>B</result>
Of the 5 distinguishable wires that lead into an apartment, 3 are for cable television service, and 3 are for telephone service. Using these wires, how many distinct combinations of 3 wires are there such that at least 1 of the wires is for cable television?