Question

How many combinations can be made up of 3 hens, 4 ducks, and 2 geese so that each combination has hens, ducks and geese? (Birds of the same kind are all different).
A. 305
B. 315
C. 320
D. 325

Answer 1

Here, we can understand the concept of combination. Also, different numbers of birds, ducks and geese will be treated as different. First find the possibilities for each and then multiply the three results to get the total number of possibilities.

Complete step by step answer:
To find at least one possibility use a formula $\left( {{2^n} - 1} \right)$, Where, $n$ is the number of given things, $n=3$ in the case of number of hens, $n=4$ in case of number of ducks and $n=2$ in case of number of geese.
Given: Three types of birds are as follows, 3 hens, 4 ducks, and 2 geese.
For at least one hen is in combination, substitute 3 for $n$ in the formula $\left( {{2^n} - 1} \right)$, $= \left( {{2^3} - 1} \right) = 8 - 1 = 7$
At least one duck is in combination, substitute 4 for $n$ in the formula $\left( {{2^n} - 1} \right)$, $= \left( {{2^4} - 1} \right) = 16 - 1 = 15$
At least one geese is in combination, substitute 2 for $n$ in the formula $\left( {{2^n} - 1} \right)$, $= \left( {{2^2} - 1} \right) = 4 - 1 = 3$
To find the total possible combinations, multiply the values of $\left( {{2^3} - 1} \right)$, $\left( {{2^4} - 1} \right)$ and $\left( {{2^2} - 1} \right)$.
Total number of possible combination $= 3 \times 15 \times 7$
Find the product to find the total number of combinations, that is $= 315$.

Hence, $315$ is the total combination that can be made. Thus, the correct option is B.

Note:
In these types of questions, find the different possibilities separately for each (in this question birds, ducks, and geese) and to get the total number of possibilities can be found out by finding the product of all possibilities. Do not perform addition here.
Always remember, wherever “and” is used in combination we do multiplication and wherever “or” is used we do addition.