Question

Amar has nine friends. He wants to invite them to his birthday party. In how many ways can he invite at least two of his friends for his birthday party?
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Answer 1

Answer:

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Answer 2

Answer:I will explain it fully so its not just you memorizing some equation. Lets focus on the pattern and start with 2 friends. The different combinations are:

a, b,ab (where “a" is the name of your first friend and “b" is the name of your second friend)

Lets focus on the distribution of numbers.

There are quantity 2 of 1 and quantity 1 of 2. 1 + 2 is 3 (in other words, there are 3 ways to invite 2 people)

a, b, c, ab, ac, bc, abc.

3 + 3 + 1 (there are 7 ways to invite 3 people)

a, b, c, d, ab, ac, ad, bc, bd, cd, abc, abd, acd, bcd, abcd

4,6,4,1 (there are 15 ways to invite 4 people)

If I list the ways they come up:

2,1

3,3 ,1

4,6,4,1

And it looks oddly like pascals triangle without the left number.

So, 7 friends should follow the line

7,21,35,35,21,7,1

The sum is 127.

Lets see if we cant figure out a way to represent this as an equation. The sum of the first in pascals series is 1. The second is 2. The 3rd, 4, the 4th is 8, 5th is 16, and so on. It looks like it doubles every time. So the seventh series is 128.

That is very close to 127. Where is that extra one?

In this case, we have excluded the instance that no friends show to the party. This is included in the original assumption “one or more".

There you have it. The number of way 7 friends can be invited to a party is 2^7 - 1 = 127.

Another way you can think about it is this:

You can either invite a friend to a party or not. This turns the function into a binary function. So a 1 next to their name means they were invited and a 0 means they werent.

One example would be:

a: 1, b: 0, c:0, d:0, e:0, f:0, g:0

In this example, only friend a comes to the party.

So your different combinations would be

00000001, 00000010, 00000011, 00000100, etc.

Again, 00000000, the situation where nobody comes does not occur. Again, it is 2^7 - 1 = 127

Step-by-step explanation: