Five cards are selected at random from a deck that contains

Five cards are selected at random from a deck that contains 32 cards. (The cards 7, 8, 9, 10, J, Q, K and Ace.) a. How many different selections, are there? b. How many different selections contain exactly two aces? c. How many different selections contain exactly two aces and three kings? d. How many different selections contain at least one ace?

Solution

Solution: This is a question of selection that is \"Combination\". Here we apply the formulae for combination.

a. Different selections: Now we have to choose five out of 32 so the formula is ³²C₅ and formula for combination is n!/r!(n-r)! so here n is 32 and r is 5. So, 32!/5!(32-5)! = 32!/5!(27)!

=32x31x30x29x28x27!/5!(27)!

27! will cut by 27!

=32x31x30x29x28/5x4x3x2x1

=201376 selecions

b. Total aces are 4 and we have to select 2 so it is ⁴C₂ and for other three cards the selection is ²⁸C_3.

Total selection = ⁴C₂x ²⁸C₃ = 4x2/2x1 x 28x27x26/3x2x1

= 3276 selections.

c. exactly two aces = ⁴C₂ and exactly three kings = ⁴C₃

  so 4x3/2x1 x 4x3x2/3x2x1

= 6 x 4 = 24 selection

d. Atleast one ace that is maximum go up to 5

So (⁴C₁ x ²⁸C₄) + (⁴C₂ x²⁸C_{3) + (}⁴C₃ x²⁸C_{2) +} (⁴C₄ x ²⁸C₁)

= 20479 + 3282 + 382 + 32

= 24175 selections