A local delivery company has three packages to deliver to three different homes. if the packages are delivered at random to the three houses, how many ways are there for at least one house to get the wrong package?
The possible outcomes that satisfy the condition of at least one house gets the wrong package are:
One house gets the wrong package or two houses get the wrong package or three houses get the wrong package.
We can calculate each of these cases and then add them together, or approach this problem from a different angle.
The only case which is left out of the condition is the case where no wrong packages are delivered.
If we determine the total number of ways the three packages can be delivered and then subtract the one case from it, the remainder will be the three cases above.
There is only one way for no wrong packages delivered to occur.
This is the same as everyone gets the right package.
The first person must get the correct package and the second person must get the correct package and the third person must get the correct package.
⇒ 1 × 1 × 1 = 1
Determine the total number of ways the three packages can be delivered.
⇒ 3 × 2 × 1 = 6
The number of ways at least one house gets the wrong package is:
⇒ 6 - 1 = 5
Therefore there are 5 ways for at least one house to get the wrong package.
How many natural numbers less than a lakh can be formed with the digits 0,6 and 9?
The digits to be used are 0,6 and 9
The required numbers are from 1 to 99999
The numbers are five digit numbers.
Therefore, every place can be filled by 0, 6 and 9 in 3 ways.
Total number of ways = 3 × 3 × 3 × 3 × 3 = 35
But 00000 is also a number formed and has to be excluded.
Total number of numbers,
= 35 - 1
= 243 - 1
There are 20 couples in a party. Every person greets every person except his or her spouse. People of the same sex shake hands and those of opposite sex greet each other with a Namaste (It means bringing one's own palms together and raising them to the chest level). What is the total number of handshakes and Namaste's in the party?
There are 20 men and 20 women.
When a man meets a woman, there are two Namastes, whereas when a man meets a man (or a woman) there is only 1 handshake.
Number of handshakes, = 2 × 20C2 (men and women ) = 2 × 190 = 380
For a number of Namastes, Every man does 19 Namastes (to the 20 women excluding his wife) and they respond in the same way.
Number of Namastes
= 2 × 20 × 19
Total number of Namastes and Handshake
= 760 + 380
In a leap year there are 366 days i.e. 52 weeks + 2 extra days.
So to have 53 Sundays one of these two days must be a Sunday.
This can occur in only 2 ways.
i.e. (Saturday and Sunday) or (Sunday and Monday).
Thus number of ways = 2
In a certain laboratory, chemicals are identified by a colour-coding system. There are 20 different chemicals. Each one is coded with either a single colour or a unique two-colour pair. If the order of colours in the pairs does not matter, what is the minimum number of different colours needed to code all 20 chemicals with either a single colour or a unique pair of colours?
Each one coded with either a single colour or unique two-colour pair.
Therefore, total number of ways = n + nC2
Minimum number of different colour needed to code all 20 chemicals will be 6
= 6 + 6C2