Question

Hey solve it by (Fundamental Principle Of Counting ) And
please send me this solution with clear handwriting image

Answer 1

this is the question related to factorial .

|_2n we can also write as 2n !

2n! = 2ⁿ. n! { 1.3.5....(2n -1) }

we know , from the basic principles of factorial .
n! =n( n -1)(n-2)(n-3).......2.1
=1.2.3.4.............(n-3)(n-2)(n-1)n ------(1)

so,
2n ! = 2n(2n -1)(2n-2)............2.1

= 1.2.3.4.........(2n -2)(2n-1)2n

= {2.4.6.8.10........(2n-2)2n}{1.3.5.7.....(2n-1)}

={ 2×2×2....n times} {1.2.3.4.5...(n-1)n } { 1.3.5.7...(2n-1)}

from equation (1)

2n != 2ⁿ. n ! { 1.3.5.7...........,.....(2n -1) }

hence proved