Question: In a group of 6 boys and 4 girls, four children are to be selected. In how many different ways can they be selected such that at least one boy should be there?
- 159
- 208
- 209
- 212
- 215
Correct Answer: C
Solution and Explanation:
Approach Solution 1:
Given that there are 6 boys and 4 girls in the group, we must determine how to choose the other four youngsters so that at least one boy will be present.
Therefore, to find a solution, you must first find the number of possibilities that you will need to choose from (6 + 5) = 10 children (no limit) to 4 children. Then find out how many possibilities a group of four children can form. Where there is no one.
Therefore, from the entire number of possible selections of the four kids, take away all the boys, giving us one boy, two boys, three boys, or four boys, which is the required response for at least one boy.
Now let us solve further
It is given 6 boys and 4 girls, in total we have 6+4=10 children and we have to select 4 children from the given total 10 children.
As a result, by employing combinations, we may argue that we can choose 4 children (at random) from a set of 10.
10c4= 10!4!(10−4)!
\(\frac{10*9*8*7}{4*3*2*1}\)
=210
Now that there are no boys present, we must choose four children from among four girls. Using a combination, we can choose four girls from among four girls in 4C4 ways, which = 1 way.
Therefore, the total number of possibilities of choosing four children without a boy
= (Number of ways to randomly select 4 children)-(Number of ways to select 4 children if there are no boys)
= 210-1 = 209
Approach Solution 2:
There is another approach to solve this question-
Let us solve this question-
It is Given: Boys = 6 Girls = 4
The formula to be used is
At least one boy = total selection - no boy selection
nCr = n!/[r!(n - r)!]
Here n is the total number possible
And r is the required selection number
Now let us solve further
Total number of children = 6 + 4 = 10
Total number of selection = 10C4 = (7 × 8 × 9 × 10)/(4 × 3 × 2) = 210
No boy selection = 4C4 = 1
Therefore at least 1 boy = 210 - 1
=209
Approach Solution 3:
Given:
Boys= 6
Girls= 4
Formula uses:
At least one boy= total selection-no boys selection
^Cr= n!/[r1!(n-r)!]
Where n= total possiable number
r= required selection number
Calculation:
Total number of children= 6+4= 10
Total number of selection
^10C4= (7*8*9*10)/(4*3*2)= 210
No boy selection
^4C4= 1
At least one boy= 210-1= 209
Therefore, there are 209 ways to selected at least one boy.
“In a group of 6 boys and 4 girls, four children are to be selected.”- is a topic of the GMAT Quantitative reasoning section of GMAT. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. GMAT Quant practice papers improve the mathematical knowledge of the candidates as it represents multiple sorts of quantitative problems.
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