(A) The first and second prizes for Mathematics can be awarded in $20 \times 19$ ways.The first and second prizes for Physics can be awarded in $20 \t

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(A) The first and second prizes for Mathematics can be awarded in $20 \times 19$ ways.The first and second prizes for Physics can be awarded in $20 \times 19$ ways.The first prize for Chemistry can be awarded in $20$ ways.The first prize for English can be awarded in $20$ ways.Since each prize is independent,the total number of ways is $20 \times 19 \times 20 \times 19 \times 20 \times 20 = 20^4 \times 19^2$.

Class 11 Mathematics Permutation and Combination Definition of permutation, Number of permutations with or without repetition, Conditional permutations

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In a class of $20$ students,in how many ways can the following prizes be awarded?
If the first and second prizes are for Mathematics,the first and second prizes for Physics,the first prize for Chemistry,and the first prize for English are to be awarded.

A
$20^4 \times 19^2$
B
$20^3 \times 19^3$
C
$20^2 \times 19^4$
D
None of these

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Permutation and Combination

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Definition of permutation, Number of permutations with or without repetition, Conditional permutations

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In a class of $20$ students,in how many ways can the following prizes be awarded?If the first and second prizes are for Mathematics,the first and second prizes for Physics,the first prize for Chemistry,and the first prize for English are to be awarded.

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Seven scientists $S_1, S_2, \ldots, S_7$ are invited to deliver one lecture each in a conference. The number of ways all the seven lectures can be arranged such that the lecture of $S_1$ is prior to that of $S_3$ and the lecture of $S_3$ is prior to that of $S_7$ is

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In a class of $20$ students,in how many ways can the following prizes be awarded?
If the first and second prizes are for Mathematics,the first and second prizes for Physics,the first prize for Chemistry,and the first prize for English are to be awarded.