$\left|\begin{array}{ccc}{ }^{\mathrm{x}}{\mathrm{C}}_{\mathrm{r}}& { }^{\mathrm{x}}{\mathrm{C}}_{\mathrm{r}+1}& { }^{\mathrm{x}}{\mathrm{C}}_{\mathrm{r}+2}\\ { }^{\mathrm{y}}{\mathrm{C}}_{\mathrm{r}}& { }^{\mathrm{y}}{\mathrm{C}}_{\mathrm{r}+1}& { }^{\mathrm{y}}{\mathrm{C}}_{\mathrm{r}+2}\\ { }^{\mathrm{z}}{C}_{\mathrm{r}}& { }^{\mathrm{z}}{\mathrm{C}}_{\mathrm{r}+1}& { }^{\mathrm{z}}{\mathrm{C}}_{\mathrm{r}+2}\end{array}\right|$ is equal to

detailed solution

Correct option is A

Δ= xCr xCr+1 xCr+2 yCr   yCr+1 yCr+2 zCr zCr+1 zCr+2          ………… (1)Applying C2→C2+C1, we getΔ= xCr x+1Cr+1 xCr+2 yCr y+1Cr+1 yCr+2 zCr z+1Cr+1 zCr+2.So, (4) is correct option.In (1), applying C3→C3+C2, we getΔ= xCr xCr+1 x+1Cr+2 yCr yCr+1 y+1Cr+2 zCr zCr+1 z+1Cr+2.So, (3) is correct option.In above applying C2→C2+C1, we getΔ= xCr x+1Cr+1 x+1Cr+2 yCr y+1Cr+1 y+1Cr+2 zCr z+1Cr+1 z+1Cr+2So, (1) is correct option.In above applying C3→C3+C2, we getΔ= xCr x+1Cr+1 x+2Cr+2 yCr y+1Cr+1 y+2Cr+2 zCr z+1Cr+1 z+2Cr+2.So, (2) is correct option.