- WELCOME TO A LESSON ON COMBINATIONS.
THE GOALS OF THIS VIDEO ARE TO EVALUATE A COMBINATION
AND ALSO TO USE COMBINATIONS TO SOLVE PROBLEMS.
A COMBINATION IS A GROUPING OR SUBSET OF ITEMS.
SO FOR A COMBINATION, THE ORDER DOES NOT MATTER.
REMEMBER FOR A PERMUTATION, THE ORDER DOES MATTER.
FOR EXAMPLE IF WE WANT TO KNOW HOW MANY COMMITTEES OF 3
CAN BE FORMED FROM A GROUP OF 4 STUDENTS,
THIS WOULD BE A COMBINATION
BECAUSE THE ORDER DOES NOT MATTER.
AND A COMBINATION CAN BE WRITTEN
USING 3 DIFFERENT TYPES OF NOTATION,
AS WE SEE HERE, HERE AND HERE.
LET'S GO AHEAD AND SEE IF WE CAN MODEL THE SITUATION.
LET'S CALL THE 4 STUDENTS "A," B, C, D,
AND LET'S SEE HOW MANY GROUPS OF 3 WE CAN FORM
USING THESE 4 LETTERS.
WELL, THE FIRST GROUP MIGHT BE "A," B, C,
AND CHANGING THE ORDER OF THESE 3 WILL NOT CREATE
A NEW COMMITTEE WHICH IS THE BIG DIFFERENCE
BETWEEN A COMBINATION AND PERMUTATION.
HERE THE ORDER DOES NOT MATTER.
SO WE COULD SELECT "A," B, D AS ANOTHER COMMITTEE OR SUBSET.
WE COULD ALSO USE B, C, D,
AND THE ONLY OTHER COMMITTEE WE COULD FORM WOULD BE "A," C, D.
SO IF WE HAVE 4 STUDENTS AND WE WANT TO FORM GROUPS OF 3,
THERE ARE ONLY 4 POSSIBLE OUTCOMES.
LET'S TAKE A LOOK AT A MATHEMATICAL WAY
OF CALCULATING A COMBINATION.
A COMBINATION OF N ITEMS TAKEN R AT A TIME
CAN BE DENOTED IN 3 DIFFERENT WAYS,
BUT = THE PERMUTATION OF N ITEMS
TAKEN R AT A TIME DIVIDED BY R FACTORIAL.
WHAT'S HAPPENING HERE IS WE'RE TAKING THE PERMUTATIONS
AND THEN WHEN WE DIVIDE BY R FACTORIAL,
WE'RE TAKING OUT THE DIFFERENT ORDERS OF THE SUBSETS
BECAUSE THE ORDER NO LONGER MATTERS,
AND THIS = N FACTORIAL DIVIDED BY N - R FACTORIAL
x R FACTORIAL.
REMEMBER THE PERMUTATION OF N ITEMS
TAKEN R AT A TIME = N FACTORIAL DIVIDED BY N - R FACTORIAL.
IF WE TAKE THIS AND DIVIDE BY R FACTORIAL,
THIS IS WHAT WE END UP WITH.
SO USING THE FORMULA, LET'S GO AHEAD AND EVALUATE 7 CHOOSE 3.
OUR NUMERATOR WOULD BE 7 FACTORIAL, THE DENOMINATOR
WOULD BE N - R FACTORIAL OR 7 - 3 FACTORIAL,
AND THEN R FACTORIAL WHICH WOULD BE 3 FACTORIAL.
LET'S GO AHEAD AND EXPAND THESE.
THIS WOULD BE 4 FACTORIAL,
SO WE'D HAVE 4, 3, 2, 1, AND THEN 3 FACTORIAL.
OF COURSE THIS SIMPLIFIES,
4 FACTORIAL GOES OUT WITH 4 FACTORIAL HERE.
3 x 2 = 6, SO WE CAN SIMPLIFY THESE 2 FACTORS WITH THE 6,
AND WE'RE LEFT WITH 7 x 5/1 = 35 COMBINATIONS.
FOR THE SECOND EXAMPLE, WE WOULD HAVE 15 FACTORIAL.
OUR DENOMINATOR WOULD BE, WELL N - R = 10,
SO WE HAVE 10 FACTORIAL AND THEN R WOULD BE 5,
SO WE HAVE 5 FACTORIAL.
NOW, LET'S AHEAD GO AND EVALUATE THIS ONE ON THE CALCULATOR.
WE CAN EVALUATE A COMBINATION IN THIS FORM
WITH A GRAPHING CALCULATOR,
BUT IT WOULD PROBABLY HELP IF WE WROTE IT
IN THIS NOTATION HERE.
SO THIS WOULD BE 15 CHOOSE 5.
AND THE REASON I SAY THAT IS TO USE THE CALCULATOR,
WE FIRST TYPE IN N WHICH IS 15, PRESS MATH.
GO OVER TO THE PROBABILITY MENU, SELECT OPTION 3 FOR COMBINATION,
AND THEN TYPE IN R WHICH IS 5.
THIS COMES OUT TO 3,003.
SO WE HAVE 3,003 COMBINATIONS IF WE HAVE 15 ITEMS
AND WE CHOOSE 5 AT A TIME.
AND JUST TO VERIFY THIS,
LET'S GO AHEAD AND EVALUATE IT IN FACTORIAL FORM AS WELL.
SO WE HAVE 15 FACTORIAL, MATH, PROBABILITY, OPTION 4,
DIVIDED BY 10 FACTORIAL x 5 FACTORIAL,
AND THE VALUE IS THE SAME AS EXPECTED.
LET'S GO AND TAKE A LOOK AT SOME PROBLEMS NOW.
THE SOCCER TEAM HAS 20 PLAYERS.
THERE ARE ALWAYS 11 PLAYERS ON THE FIELD.
HOW MANY DIFFERENT GROUPS OF PLAYERS CAN BE ON THE FIELD
AT ANY ONE TIME?
SO WE'RE NOT CONSIDERING THEIR POSITIONS,
SO THE ORDER DOES NOT MATTER, THEREFORE THIS IS A COMBINATION.
SO WE HAVE A COMBINATION OF 20 PLAYERS TAKEN 11 AT A TIME.
WE CAN WRITE IT LIKE THIS, LIKE THIS, OR EVEN LIKE THIS.
SO THIS = 20 FACTORIAL DIVIDED BY 20 - 11 FACTORIAL,
THAT WOULD BE 9 FACTORIAL,
AND THEN R FACTORIAL WHICH IS 11 FACTORIAL.
NOW AGAIN, WE COULD DO THIS BY HAND,
BUT I'M GOING TO GO AHEAD AND USE THE CALCULATOR
DUE TO TIME.
AND I WILL USE THE COMBINATION FEATURE OF THE CALCULATOR,
SO WE'LL PRESS 20, MATH, PROBABILITY, OPTION 3,
AND THEN PRESS R WHICH IS 11, ENTER.
SO WE HAVE 167,000,960 POSSIBLE COMBINATIONS OF PLAYERS
ON THE FIELD.
HERE A STUDENT NEEDS 8 MORE CLASSES TO COMPLETE HER DEGREE.
IF SHE HAS MET THE REQUISITES FOR ALL OF THE COURSES,
HOW MANY WAYS CAN SHE TAKE FOUR CLASSES NEXT SEMESTER?
AGAIN, THE ORDER DOES NOT MATTER,
THEREFORE THIS WOULD BE A COMBINATION
OF 8 ITEMS CHOSEN 4 AT A TIME.
LET'S GO AHEAD AND DO THIS ONE BY HAND.
SO WE WOULD HAVE 8 FACTORIAL.
OUR DENOMINATOR WOULD BE 8 - 4 FACTORIAL, THAT'S 4 FACTORIAL,
AND THEN R FACTORIAL WHICH IS ALSO 4 FACTORIAL,
SO WE'D HAVE 8 x 7 x 6 x 5.
I'M GOING TO STOP HERE AND WRITE 4 FACTORIAL.
AND FOR THE DENOMINATOR,
WE'LL HAVE 4 x 3 x 2 x 1 x THIS FACTORIAL AS WELL.
THAT SAVES A LITTLE BIT OF WRITING,
BECAUSE THE 4 FACTORIAL SIMPLIFY OUT.
LET'S SEE WHAT ELSE WE CAN DO.
WELL, HERE'S A FACTOR OF 4 AND 2, THAT WOULD BE 8,
SO WE CAN SIMPLIFY THOSE.
WE'RE LEFT WITH 3.
WELL, HERE'S A 6, SO WE CAN SIMPLIFY THE 3 OUT
AND CHANGE THE 6 TO A 2.
SO WE HAVE 7 x 2 x 5 = 70.
SO GIVEN 8 CLASSES, THERE ARE 70 WAYS
THAT SHE CAN CHOOSE 4 OF THEM.
LET'S GO AHEAD AND VERIFY THIS WITH THE CALCULATOR.
SO WE'LL HAVE 8 CHOOSE 4, OPTION 3, PRESS 4 FOR R,
AND THERE WE GO.
LET'S TAKE A LOOK AT ONE MORE.
THERE ARE 4 MEN AND 5 WOMEN IN A SMALL OFFICE.
THE CUSTOMER WANTS A SITE VISIT
FROM A GROUP OF 2 MEN AND 2 WOMEN.
HOW MANY DIFFERENT GROUPS CAN BE FORMED FROM THE OFFICE?
WELL AGAIN, THE ORDER DOESN'T MATTER
BECAUSE WE'RE FORMING GROUPS,
BUT THERE ARE REALLY 2 EVENTS GOING ON HERE.
WE HAVE TO SELECT OR CHOOSE THE MEN,
AND THEN WE ALSO HAVE TO CHOOSE OR SELECT THE WOMEN.
SO WE'RE GOING TO FIND THE NUMBER OF WAYS
THAT WE CAN SELECT THE MEN,
AND WE'RE GOING TO MULTIPLY IT BY THE NUMBER OF WAYS
THAT WE CAN SELECT THE WOMEN.
SO WE'LL FIRST CONSIDER THE NUMBER OF WAYS
THAT WE CAN CHOOSE THE MEN.
WELL, THERE ARE 4 MEN AND WE'RE CHOOSING 2 AT A TIME,
SO THAT WOULD BE 4 CHOOSE 2.
AND THEN FOR THE WOMEN, THERE'S 5 TO CHOOSE FROM,
SO THIS WOULD BE x 5 CHOOSE 2.
SO LET'S GO AHEAD AND WRITE THIS OUT.
WE'D HAVE 4 FACTORIAL/4 - 2 FACTORIAL, THAT'S 2 FACTORIAL.
THEN ANOTHER 2 FACTORIAL FOR R x,
HERE WE'D HAVE 5 FACTORIAL/5 - 2 FACTORIAL,
THAT'S 3 FACTORIAL x 2 FACTORIAL.
SO THERE'S OUR NUMERATOR,
AND OUR DENOMINATOR WOULD LOOK LIKE THIS.
NOW, LET'S SIMPLIFY.
WELL, HERE'S A 2 x A 2, THAT WOULD SIMPLIFY WITH THE 4.
HERE WE HAVE 3 x 2 x 1, THAT WOULD SIMPLIFY HERE AND HERE.
AND THEN LASTLY, WE HAVE THIS FACTOR OF 2
WHICH WOULD SIMPLIFY WITH THIS 4, THAT WOULD CHANGE TO 1,
THAT WOULD CHANGE TO A 2.
SO IT LOOKS LIKE WE HAVE 6 x 10 = 60.
SO THEY COULD FORM 60 DIFFERENT GROUPS OF 2 MEN AND 2 WOMEN
FROM 4 MEN AND 5 WOMEN FOR THE SITE VISIT.
OKAY, THANK YOU FOR WATCHING,
AND I HOPE YOU FOUND THIS HELPFUL.