Question

a gambler began playing blackjack with $110 in chip.After exactly 12 hands, he left the table with $320 in chip, having won some hands and lost others each win earns 100 and each loss cost 10 ( dollars). haw many possible outcomes were there for the first 5 hands he played?​

Answer 1

Given:

The gambler:

$\implies$ Started with $110

$\implies$ Left with $320

Thus, he in 12 hands won:

$\boxed{\sf{320 - 110 = 210}}$

That is:

$\implies$ 100W - 10L = 210

$\implies$ 100W - 10(12-W) = 210

Note: Wins + Loss = 12

That is: W = 3

So:

Out of 12 hands, the gambler won 3 hands and lost 9.

Now:

For the first 5 hands played, there could be the following outcomes:

(i) WWWLL = 5!/(3!2!) = 10 ways

For example:

WWWLL, WWLWL, WLWWL, ... And so on.

(ii) WWLLL = 5!/(3!2!) = 10 ways

(iii) WLLLL = 5!/(4!1!) = 5 ways

(iv) LLLLL = Only 1 way

Total:

= 10 + 10 + 5 + 1

= 20 + 6

= 26

Final answer: 26 ways