1. 6.3 Find Probabilities Using Combinations

2. What is a combination?
A combination is a selection of objects in which order is NOT important.
The number of combinations of n objects taken r at a time, where r < n, is given by:

3. Count Combinations
One way of finding the number of combinations is by counting.
List all the permutations and cross out any duplicate groupings.

4. Example: Counting Combinations
Count the combinations of 2 letters taken from the list of letters M, A, T, H, E, M, A, T, I, C, S.
MA, MT, MH, ME, MM, MA, MT, MI, MC, MS
AM, AT, AH, AE, AM, AA, AT, AI, AC, AS
TM, TA, TH, TE, TM, TA, TT, TI, TC, TS
HM, HA, HT, HE, HM, HA, HT, HI, HC, HS
EM, EA, ET, EH, EM, EA, ET, EI, EC, ES
MM, MA, MT, MH, ME, MA, MT, MI, MC, MS
AM, AA, AT, AH, AE, AM, AT, AI, AC, AS
TM, TA, TT, TH, TE, TM, TA, TI, TC, TS
IM, IA, IT, IH, IE, IM, IA, IT, IC, IS
CM, CA, CT, CH, CE, CM, CA, CT, CI, CS
SM, SA, ST, SH, SE, SM, SA, ST, SI, SC

5. Cross out any duplicate groupings.
MA, MT, MH, ME, MM, MA, MT, MI, MC, MS
AM, AT, AH, AE, AM, AA, AT, AI, AC, AS
TM, TA, TH, TE, TM, TA, TT, TI, TC, TS
HM, HA, HT, HE, HM, HA, HT, HI, HC, HS
EM, EA, ET, EH, EM, EA, ET, EI, EC, ES
MM, MA, MT, MH, ME, MA, MT, MI, MC, MS
AM, AA, AT, AH, AE, AM, AT, AI, AC, AS
TM, TA, TT, TH, TE, TM, TA, TI, TC, TS
IM, IA, IT, IH, IE, IM, IA, IT, IC, IS
CM, CA, CT, CH, CE, CM, CA, CT, CI, CS
SM, SA, ST, SH, SE, SM, SA, ST, SI, SC

6. How many groupings are possible?
The groupings left that do not duplicate are: MA, MT, MH, ME, MM, MI, MC, MS, AT, AH, AE, AA, AI, AC, AS, TH, TE, TT, TI, TC, TS, HE, HI, HC, HS, EI, EC, ES, IC, IS, CS
There are 31 possible combinations of 2 letters from the list M, A, T, H, E, M, A, T, I, C, S.

7. Use the Combinations Formula
It is much easier to use the combination formula to find the possible combinations.

8. Find a Probability using Combinations
There are 16 students in your class. The teacher chooses 4 students at random to complete a project. Find the probability that you and your best friend will be in the same group.
Find the total number of possible groups first.

9. If you and your friend are chosen together, there are 2 other students to be chosen from the remaining 14 to complete your group.
The number of favorable outcomes is as follows:
There are 91 favorable outcomes.
The probability you and your best friend are in the same group is