Question

Ramesh has 6 friends. In how many ways can he invite one
or more of them at a dinner?
(a) 61 (b) 62 (c) 63 (d) 64

Answer 1

Given:

Number of friends of Ramesh = 6

We have to find, in how many ways can he invite one  or more of them at a dinner.

Solution:

∴ Total number of ways can he invite one  or more of them at a dinner

= $^6C_{1} +^6C_{2} +^6C_{3} +^6C_{4} +^6C_{5} +^6C_{6}$

Using the formula:

$^nC_{r} =\dfrac{n!}{r!(n-r)!}$

= $\dfrac{6!}{1!(6-1)!} +\dfrac{6!}{2!(6-2)!}+\dfrac{6!}{3!(6-3)!}+\dfrac{6!}{4!(6-4)!}+\dfrac{6!}{5!(6-5)!}+\dfrac{6!}{6!(6-6)!}$

= $\dfrac{5!.6}{(5)!} +\dfrac{4!.5.6}{2(4)!}+\dfrac{3!.4.5.6}{6(3)!}+\dfrac{4!.5.6}{4!(2)}+\dfrac{6!}{5!(1)!}+\dfrac{6!}{6!(0)!}$

= 6 + 15 + 20 + 15 + 6 + 1

= 63

∴ Total number of ways can he invite one  or more of them at a dinner = 63

Thus, the required "option is C) 63".

Answer 2

Given:

Ramesh has 6 friends.

To find:

In how many ways can he invite one  or more of them at a dinner?

Solution:

From the given information, we have the data as follows.

Ramesh has 6 friends.

The number of ways he can invite one  or more of them at dinner is calculated as follows.

Let us consider each case one by one.

If Ramesh invites only 1 friend, that can be done in ⁶C₁ = 6

If Ramesh invites only 2 friends, then ⁶C₂ = 15

If Ramesh invites only 3 friends, then  ⁶C₃ = 20

If Ramesh invites only 4 friends, then  ⁶C₄ = 15

If Ramesh invites only 5 friends, then  ⁶C₅ = 6

If Ramesh invites all his friends, then  ⁶C₆ = 1

Summation of the above is,

= ⁶C₁ + ⁶C₂ + ⁶C₂ + ⁶C₄ + ⁶C₅ + ⁶C₆

= 6 + 15 + 20 + 15 + 6 + 1

= 63

Therefore, in 63 ways Ramesh can invite one  or more of them at a dinner.