1.

Evaluate `|{:(""^(m)C_(1),""^(m)C_(2),""^(m)C_(3)),(""^(n)C_(1),""^(n)C_(2),""^(n)C_(3)),(""^(p)C_(1),""^(p)C_(2), ""^(p)C_(3)):}|`

Answer» Let the given determinant be `Delta`. Then,
`Delta = |{:(m, (1)/(2)m(m-1),(1)/(6)*m(m-1)(m-2)),(n, (1)/(2)n(n-1),(1)/(6)*n(n-1)(n-2)),(p,(1)/(2)p(p-1),(1)/(6)*p(p-1)(p-2)):}|`
`=((1)/(2) xx (1)/(6) xx mnp)* |{:(1,(m-1),(m-1)(m-2)),(1,(n-1),(n-1)(n-2)),(1,(p-1),(p-1)(p-2)):}|`
`=(1)/(12)*mnp* |{:(1,m-1,(m-1)(m-2)),(0,n-m,(n-m)(n+m-3)),(0,p-m,(p-m)(p+m-3)):}| [R_(2) to (R_(2)-R_(1)) "and "R_(3) to (R_(3) -R_(1))]`
`=(1)/(12)*(mnp)(n-m)(p-m)* |{:(1,m-1,(m-1)(m-2)),(0," "1,(n+m-3)),(0," "1,(p+m-3)):}|`
`["taking (n-m) common from "R_(2) "and (p-m) common from"R_(3)]`
` =(1)/(12) * (mnp)(n-m)(p-m) *1*|{:(1, (n+m-3)), (1, (p+m-3)):}|`
`=(1)/(12) * (mnp)(n-m)(p-m)[(p + m-3)-(n+m-3)]`
`=(1)/(12) *(mnp)(n-m)(p-m)(p-n)`
`=(1)/(12) * (mnp)(m-n)(n-p)(p-m)`
Hence, `Delta = (1)/(12) * (mnp)(m-n)(n-p)(p-m)`


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