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  • In a lottery game, you need to select 5 numbers from a pool of 30. How many different combinations of numbers are possible if one number must be odd?  <stron

In a lottery game, you need to select 5 numbers from a pool of 30. How many different combinations of numbers are possible if one number must be odd?
 

Option: 1

291652


Option: 2

310953


Option: 3

610520


Option: 4

712650


Answers (1)

best_answer

To calculate the number of different combinations of 5 numbers that can be selected from a pool of 30 , with one number being odd, we can consider two cases: selecting one odd number and four additional numbers, and selecting all five numbers as odd.

Case 1: Selecting one odd number and four additional numbers:
Selecting one odd number: There are 15 odd numbers in the pool of 30 .
Selecting four additional numbers: There are 29 numbers remaining after selecting the odd number, and we need to choose 4 numbers from this pool.
Therefore, the total number of combinations for Case 1 is

\mathrm{C}(15,1) \times \mathrm{C}(29,4).

Case 2: Selecting all five numbers as odd:
Selecting all five numbers: There are 15 odd numbers in the pool of 30 , and we need to choose 5 numbers from this pool.

Therefore, the total number of combinations for Case 2 is \mathrm{C}(15,5).
To find the total number of different combinations, we sum up the results of both cases:

Total combinations =\mathrm{C}(15,1) \times \mathrm{C}(29,4)+\mathrm{C}(15,5)

Calculating these values, we get:

\begin{aligned} & \mathrm{C}(15,1)=15 \\ & \mathrm{C}(29,4)=\frac{29 ! }{(4 !(29-4) !)}==20,475 \\ & \mathrm{C}(15,5)=\frac{15 !}{(5 !(15-5) !)}=\frac{15 ! }{(5 ! 10 !)}==3,003 \end{aligned}

Total combinations = 15 \times 20,475+3,003=307,950+3,003=310,953

Therefore, there are 310,953 different combinations of 5 numbers that can be selected from a pool of 30 , with one number being odd, in the lottery game.

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