Question

In a hospital a trial of 'RT-PCR' test is conducted for covid test and they want to perform as many trials as they can out of $40$ patients by calling the same number of patients but not the same group. Then

• A. maximum number of trials can be performed is ${}^{39}{C}_{19}$
• B. maximum number of trials can be performed is ${}^{40}{C}_{20}$
• C. the number of times each patient can be tested, when maximum number of trials can be performed is ${}^{40}{C}_{20}$
• D. the number of times each patient can be tested, when maximum number of trials can be performed is ${}^{39}{C}_{20}$

Answer 1

Answer: (B), (D)

the number of times each patient can be tested, when maximum number of trials can be performed is ${}^{39}{C}_{20}$

Let a trial of $r$ patients is conducted out of $40$ patients.
Case $1:$
Number of trials conducted $=$ Number of such combinations $={}^{40}{C}_r$
$\Rightarrow {}^n{C}_r$ is maximum when $r=\frac{n}{2}=\frac{40}{2}=20$
$\therefore$ Maximum value of ${}^{40}{C}_r={}^{40}{C}_{20}$
$\Rightarrow$ Maximum trials can be performed $={}^{40}{C}_{20}$

Case $2:$ When a maximum number of trials performed a group of $20$ patients should be made.
$\therefore$ One patient can be grouped with other $19$ patients out of $39$.
$={}^{40-1}{C}_{20-1}={}^{39}{C}_{19}={}^{39}{C}_{20}$