The other answer given is ("fortunately") "completely" incorrect. Here's why;
The chance for a Pokemon to be shiny is 1/4096 (3/4096 if you have the Shiny Charm).
In a horde, there are 5 Pokemon, you have to consider 3 to be shiny and 2 to be non-shiny. That means you get 1/4096 1/4096 1/4096 4095/4096 4095/4096 = 16,769,025/1,152,921,504,606,846,976 = 1,4545 10^-11.
However, you have to consider picking them in a different order (if you had only two Pokemon and ask for one to be shiny, the orders AB and BA wouldbe equal for you ("one shiny"), but they are different possibilities nonetheless). For that reason, you have to multiply that number by 10 (different ordering possibilities - if shinies are A and non-shinies B, then all of those would be AAABB, AABAB, ABAAB, BAAAB, AABBA, ABABA, BAABA, ABBAA, BABAA, BBAAA <- 10 possibilities), so ultimately, you'd get 167,690,250/1,152,921,504,606,846,976, or roughly 1,4545 10^-10. In percents, that's *1,4545 10^-8 %, or 0,000000014545 %.**
Now, with the Shiny Charm, the game generates another 2 codes trying for a shinies, which effectively means the chance of 3/4096 for a shiny. The calculation will be the same, except with the chance 3/4096 for shiny Pokemon and 4093/4096 for non-shiny Pokemon. The results you get is 4,523,215,230/1,152,921,504,606,846,976 which is equal to *3.9233 10^-9 % or 0,00000039233 %**.
