For combinations we could use this homemade function

combination = function(size,sample) {
  return(factorial(size)/(factorial(sample)*factorial(size-sample)))
}
combination(83,4)
## [1] 1837620

but lets just use R’s built in Choose function.

choose(83,4)
## [1] 1837620

For binomial distribution, we can use R’s dbinom function or the following formula.

(a)

choose(6,3)
## [1] 20

(b)

#dbinom(size = 5,prob = .2,4)
choose(5,4)*(.2)^4*(.8)^1
## [1] 0.0064

(c)

choose(7,2)
## [1] 21

(d)

choose(26,26)
## [1] 1

(e)

choose(4,3)*(.2)^3*(.8)^1
## [1] 0.0256

(f)

choose(6,2)
## [1] 15

(g)

choose(10,9)
## [1] 10

(h)

choose(8,5)*(.3)^5*(.7)^3
## [1] 0.04667544