Question

Let C(n, k) = the number of k-membered subsets of an n-membered set. Find (a) C(6, k) for k = 0,1,2,...,6 (b) C(7, k) for k = 0,1.2...,7

Answer 1

(a)
`C(6,0) = 1`,
`C(6,1) = 6`,
`C(6,2) = 15`,
`C(6,3) = 20`,
`C(6,4) = 15`,
`C(6,5) = 6`,
`C(6,6) = 1`.

(b)
`C(7,0) = 1`,
`C(7,1) = 7`,
`C(7,2) = 21`,
`C(7,3) = 35`,
`C(7,4) = 35`,
`C(7,5) = 21`,
`C(7,6) = 7`,
`C(7,7)=1`.

Explanation:

In this exercise we only need to recall the formula for C(n,k):

`C(n,k) = (n!)/(k!(n-k)!)`

where the symbol
`n!` is the factorial and means

`n! = 1\cdot 2\cdot 3\cdot 4\cdtos (n-1)\cdot n`.

By convention 0!=1. The most important property of the factorial is
`n!=(n-1)!\cdot n`, for example 3!=1*2*3=6.

(a) The explanations to the solutions is just the calculations.

• 
  `C(6,0) = (6!)/(0!(6-0)!) = (6!)/(6!) = 1`

• 
  `C(6,1) = (6!)/(1!(6-1)!) = (6!)/(5!) = (5!\cdot 6)/(5!) = 6`

• 
  `C(6,2) = (6!)/(2!(6-2)!) = (6!)/(2\cdot 4!) = (5!\cdot 6)/(2\cdot 4!) = (4!\cdot 5\cdot 6)/(2\cdot 4!) = (5\cdot 6)/(2) = 15`

• 
  `C(6,3) = (6!)/(3!(6-3)!) = (6!)/(3!\cdot 3!) = (5!\cdot 6)/(6\cdot 6) = (5!)/(6) = (120)/(6) = 20`

• 
  `C(6,4) = (6!)/(4!(6-4)!) = (6!)/(4!\cdot 2!) = frac{5!\cdot 6}{2\cdot 4!} = (4!\cdot 5\cdot 6)/(2\cdot 4!) = (5\cdot 6)/(2) = 15`

• 
  `C(6,5) = (6!)/(5!(6-5)!) = (6!)/(5!) = (5!\cdot 6)/(5!) = 6`

• 
  `C(6,6) = (6!)/(6!(6-6)!) = (6!)/(6!) = 1`.

(b) The explanations to the solutions is just the calculations.

• 
  `C(7,0) = (7!)/(0!(7-0)!) = (7!)/(7!) = 1`

• 
  `C(7,1) = (7!)/(1!(7-1)!) = (7!)/(6!) = (6!\cdot 7)/(6!) = 7`

• 
  `C(7,2) = (7!)/(2!(7-2)!) = (7!)/(2\cdot 5!) = (6!\cdot 7)/(2\cdot 5!) = (5!\cdot 6\cdot 7)/(2\cdot 5!) = (6\cdot 7)/(2) = 21`

• 
  `C(7,3) = (7!)/(3!(7-3)!) = (7!)/(3!\cdot 4!) = (6!\cdot 7)/(6\cdot 4!) = (5!\cdot 6\cdot 7)/(6\cdot 4!) = (120\cdot 7)/(24) = 35`

• 
  `C(7,4) = (7!)/(4!(7-4)!) = (6!\cdot 7)/(4!\cdot 3!) = frac{5!\cdot 6\cdot 7}{4!\cdot 6} = (120\cdot 7)/(24) = 35`

• 
  `C(7,5) = (7!)/(5!(7-2)!) = (7!)/(5!\cdot 2!) = 21`

• 
  `C(7,6) = (7!)/(6!(7-6)!) = (7!)/(6!) = (6!\cdot 7)/(6!) = 7`

• 
  `C(7,7) = (7!)/(7!(7-7)!) = (7!)/(7!) = 1`

For all the calculations just recall that 4! =24 and 5!=120.