Let ar=r 4Cr , br=(4 - r)( )4Cr , Ar=[ ar 2 3 br ] and A= displaystyle ∑ r = 04Ar , then the value of |A| is equal to

Question

Let ar=r 4Cr , br=(4 - r)( )4Cr , Ar=[ ar 2 3 br ] and A= displaystyle ∑ r = 04Ar , then the value of |A| is equal to (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

A= displaystyle ∑ r = 04Ar=[ displaystyle ∑ r = 04r4Cr displaystyle ∑ r = 042 displaystyle ∑ r = 043 displaystyle ∑ r = 04(4 - r)4Cr ] Now, displaystyle ∑ r = 04r4Cr= displaystyle ∑ r = 04(4 - r)4Cr=4× 23=25 A=[ 32 10 15 32 ] |A|=1024-150=874

Q. Let $a_{r} = r$ $^{4} C_{r}$ , $b_{r} = (4 - r) ()^{4} C_{r}$ , $A_{r} = [a_{r} 3 2 b_{r}]$ and $A = r = 0 \sum 4 A_{r}$ , then the value of $∣ A ∣$ is equal to

$A = r = 0 \sum 4 A_{r} = ⎣ ⎡ r = 0 \sum 4 r^{4} C_{r} r = 0 \sum 4 3 r = 0 \sum 4 2 r = 0 \sum 4 (4 - r)^{4} C_{r} ⎦ ⎤$
Now, $r = 0 \sum 4 r^{4} C_{r} = r = 0 \sum 4 (4 - r)^{4} C_{r} = 4 \times 2^{3} = 2^{5}$
$A = [32151032]$
$∣ A ∣ = 1024 - 150 = 874$

Q. Let ar​=r 4Cr​ , br​=(4−r)()4Cr​ , Ar​=[ar​3​2br​​] and A=r=0∑4​Ar​ , then the value of ∣A∣ is equal to

A=r=0∑4​Ar​=⎣⎡​r=0∑4​r4Cr​r=0∑4​3​r=0∑4​2r=0∑4​(4−r)4Cr​​⎦⎤​ Now, r=0∑4​r4Cr​=r=0∑4​(4−r)4Cr​=4×23=25 A=[3215​1032​] ∣A∣=1024−150=874

Q. Let $a_{r} = r$ $^{4} C_{r}$ , $b_{r} = (4 - r) ()^{4} C_{r}$ , $A_{r} = [a_{r} 3 2 b_{r}]$ and $A = r = 0 \sum 4 A_{r}$ , then the value of $∣ A ∣$ is equal to

$A = r = 0 \sum 4 A_{r} = ⎣ ⎡ r = 0 \sum 4 r^{4} C_{r} r = 0 \sum 4 3 r = 0 \sum 4 2 r = 0 \sum 4 (4 - r)^{4} C_{r} ⎦ ⎤$
Now, $r = 0 \sum 4 r^{4} C_{r} = r = 0 \sum 4 (4 - r)^{4} C_{r} = 4 \times 2^{3} = 2^{5}$
$A = [32151032]$
$∣ A ∣ = 1024 - 150 = 874$