Directions (48-51): The scatter chart given below shows the number of boys on (Y – axis) and number of girls (X – axis) in three different class of P, Q and R.

NOTE: If there were 10 more girls in class R then the probability of selecting a girl from R would be 40%

The scatter chart given below shows the number of boys on (Y – axis) and number of girls (X – axis) in three different class of P, Q and R.

NOTE: If there were 10 more girls in class R then the probability of selecting a girl from R would be 40%

Number of girls in class P is 40% of the number of girls in R. If a teacher wants to make a two groups (A and B) from class R and P respectively such that in group A two girls and one boy are selected and in group B two girls and one boy are slected. Find the difference between the number of ways he can make such group from each class.

Based on the provided scatter chart, we can determine the number of boys and girls in classes P and R to calculate the number of ways to form the groups.

Step 1: Extracting values from the scatter chart
From the scatter chart:
- The Y-axis represents the number of boys.
- The X-axis represents the number of girls.
For class P:
- Number of boys = 40
- Number of girls = a + b

For class R:
- Number of boys = 60
- Number of girls = a + 5b

Step 2: Finding the number of girls in class R
According to the note, if there were 10 more girls in class R, the probability of selecting a girl from class R would be 40% (or 0.4).
Let $G_R$ be the number of girls in class R.
The probability equation is:

$\frac{G_R+10}{G_R+10+60}=0.4$

$\frac{G_R+10}{G_R+70}=\frac{2}{5}$

$5(G_R+10)=2(G_R+70)$

$5G_R+50=2G_R+140$

$3G_R=90$

$G_R=30$

Step 3: Finding the number of girls in class P
The problem states that the number of girls in class P is 40% of the number of girls in R.
Let $G_P$ be the number of girls in class P.

$G_P=40\%of30=0.4\times 30=12$

Step 4: Calculating the number of ways to form the groups
Group A (from class R):
We need to select 2 girls from 30 girls, and 1 boy from 60 boys.
Number of ways to form Group A:

$\text{Ways}\left(A\right)={ }^{30}{C}_2\times { }^{60}{C}_1$

${ }^{30}{C}_2=\frac{30\times 29}{2}=435$

${ }^{60}{C}_1=60$

$\text{Ways}\left(A\right)=435\times 60=26100$

$\text{Ways}\left(B\right)={ }^{12}{C}_2\times { }^{40}{C}_1$

${ }^{12}{C}_2=\frac{12\times 11}{2}=66$

${ }^{40}{C}_1=40$

$\text{Ways}\left(B\right)=66\times 40=2640$

Step 5: Calculating the difference between the number of ways

$\text{Difference}=\text{Ways}\left(A\right)-\text{Ways}\left(B\right)$

$\text{Difference}=26100-2640=23460$

Therefore, the difference between the number of ways to make the groups is 23460.