SOLUTION: Kate wants to choose 3 out 13 different items as the christmas presents. Given that 4 items are from shop B, and 3 items from shop C, find the number of combinations in each of the

Algebra ->  Permutations -> SOLUTION: Kate wants to choose 3 out 13 different items as the christmas presents. Given that 4 items are from shop B, and 3 items from shop C, find the number of combinations in each of the      Log On


   

Question 1173585: Kate wants to choose 3 out 13 different items as the christmas presents. Given that 4 items are from shop B, and 3 items from shop C, find the number of combinations in each of the following cases.
A. Choosing 3 items from the same shop.
B. Choosing 3 items which are not from shop A
C. Choosing at least 1 item from shop A
Found 2 solutions by ewatrrr, ikleyn:
Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!
 

Hi,

13 Items:  6 ShopA   4 ShopB     3 ShopC

Choosing 3, find the number of combinations in each of the following cases.

nCr = (n!)/(r!(n - r)!).   nCr on Your Caclulator

A. Choosing 3   items from the same shop: 6C3 + 4C3 + 3C3 =   20+4+1 = 25

B. Choosing 3 items which are not from shop A:  7C3 = 35

C. Choosing at least 1 item from shop A:  

  (6C1)(7C2) + (6C2)(7C1) + (6C3) = (6)(21) + (15)(7) + (20)

Wish You the Best in your Studies.

Answer by ikleyn(52776) About Me  (Show Source):
You can put this solution on YOUR website!
.
Kate wants to choose 3 out 13 different items as the christmas presents.
Given that 4 items are from shop B, and 3 items from shop C,
find the number of combinations in each of the following cases.
A. Choosing 3 items from the same shop.
B. Choosing 3 items which are not from shop A
C. Choosing at least 1 item from shop A
~~~~~~~~~~~~~~~


            In the post by  @ewatrrr,  part  (A)  and part (B)  answered correctly,  but part  (C)  answered  INCORRECTLY.

            I came to bring the correct solution to part  (C). 


So we have a pool of  13 items, in all;

    6 of them are from shop A;   4 are from shop B   and   3 are from shop C.


(C).  Find the number of combinations, given that at least 1 item is from shop A.


      Then we have THESE 3 possible cases


          1 item is from A:     C%5B6%5D%5E1%2AC%5B7%5D%5E2 = 6*21 = 126 combinations


          2 items are from A:   C%5B6%5D%5E2%2AC%5B7%5D%5E1 = 15*7 = 105 combinations

          
          3 items are from A:   C%5B6%5D%5E3 = 20 combinations.


     The total of these cases is  C%5B6%5D%5E1%2AC%5B7%5D%5E2 + C%5B6%5D%5E2%2AC%5B7%5D%5E1 + C%5B6%5D%5E3 = 126 + 105 + 20 = 251.   ANSWER for part (C)

Solved.


. . . . . . . .

@ewatrrr fixed her post after seeing my solution.