Question

Gin Rummy and are holding 10 cards in your hand from a standarddeck of 52 cards. Find the probability t

Answer 1

Here is the correct question

Gin Rummy and are holding 10 cards in your hand from a standard deck of 52 cards.

a) How many ways can you select three kings and four 6's from the deck, the remaining cards not kings or 6's

b) Find the probability that three kings and four 6's are being held by you from the deck.

Answer:

a) 52976

b) $\mathbf{3.34866744 \times 10^{-6}}$

Step-by-step explanation:

So from the information above,

no of ways three kings can be selected and four 6's from the deck without the remaining cards being kings or 6's can be expressed as:

no of ways to select 3 kings from 4 kings is $(^4_3)$

no of ways to select 4 6's from (4 6's) is $(^4_4)$

no of ways to select the remaining 3 cards from (52 - 4 - 4) = 44 cards is $(^{44}_3)$

Mathematically, multiplying the three together we have:

$= (^4_3) (^4_4) (^{44}_3)$

$= \dfrac{4!}{3!(4-3)!} \times \dfrac{4!}{4!(4-4)!} \times \dfrac{44!}{3!(44-3)!}$

$= 4 \times 1 \times 13244$

= 52976

To find the probability that you are holding three kings and four 6's from the deck, the remaining cards not kings or 6's can be calculated as follows;

To do this, we need to first know the no of ways we can  select 10 cards out of the pack of 52 cards

i.e

$(^{52}_{10})$

= $\dfrac{52!}{10!(52-10)!}$

= $\dfrac{52!}{10!(42)!}$

= 1.58200242 × 10¹⁰

Now, to find the  probability that you are holding three kings and four 6's from the deck, the remaining cards not kings or 6's :

we will need to divide the number of ways we can select three kings and four 6's from the deck by the no of ways we can select 10 cards out of the pack of 52 cards.

Mathematically; we have,

$=\dfrac{52976}{1.58200242 \times 10^{10}}$

= $\mathbf{3.34866744 \times 10^{-6}}$