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Question

∣ ∣xCrxCr+1xCr+2yCryCr+1yCr+2zCrzCr+1zCr+2∣ ∣=∣ ∣ ∣xCrx+kCr+kx+mCr+myCry+pCr+py+sCr+szCrz+lCr+lz+tCr+t∣ ∣ ∣.Find k+m+p+s+l+t?

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Solution

L.H.S =∣ ∣xCrxCr+1xCr+2yCryCr+1yCr+2zCrzCr+1zCr+2∣ ∣
Applying C2C2+C1 and C3C3+C2
∣ ∣xCrxCr+1+xCrxCr+2+xCr+1yCryCr+1+yCryCr+2+yCr+1zCrzCr+1+zCrzCr+2+zCr+1∣ ∣
∣ ∣ ∣xCrx+1Cr+1x+1Cr+2yCry+1Cr+1y+1Cr+2zCrz+1Cr+1z+1Cr+2∣ ∣ ∣ (By Pascal
triangle rub i.e., nCr+nCr1=n+1Cr)
Applying C3C3+C2
=∣ ∣ ∣xCrx+1Cr+1x+1Cr+1+x+1Cr+2yCry+1Cr+1y+1Cr+1+y+1Cr+2zCrz+1Cr+1z+1Cr+1+z+1Cr+2∣ ∣ ∣
=∣ ∣ ∣xCrx+1Cr+1x+2Cr+2yCry+1Cr+1y+2Cr+2zCrz+1Cr+1z+2Cr+2∣ ∣ ∣

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