The value of ∑ _ r = 0 ^ 6 (Cr 6 · C6 - r 6) is equal to :

We have $\mathop{∑}_{r = 0}^{6}\left(Cr 6 \cdot C6 - r 6\right) = C0 6 \cdot C6 6 + C1 6 \cdot C5 6 + . . . + C6 6 \cdot C0 6$

Now, we know that the expansion of ${\left(1 + x\right)}^{n} = C0 n {x}^{0} + C1 n {x}^{1} + C2 n {x}^{2} + . . . Cn n {x}^{n}$

$⇒ ( 1 + x{)}^{6} ( 1 + x{)}^{6} = \left(C0 6 + C1 6 x + C2 6 {x}^{2} + . . . + C6 6 {x}^{6}\right)\left(C0 6 + C1 6 x + C2 6 {x}^{2} + . . . + C6 6 {x}^{6}\right)$

$⇒ ( 1 + x{)}^{12} = \left(C0 6 + C1 6 x + C2 6 {x}^{2} + . . . + C6 6 {x}^{6}\right)\left(C0 6 + C1 6 x + C2 6 {x}^{2} + . . . + C6 6 {x}^{6}\right)$

On comparing the coefficient of ${x}^{6}$ both sides, we get

$C6 12 = C0 6 \cdot C6 6 + C1 6 \cdot C5 6 + . . . + C6 6 \cdot C0 6$

$⇒ C0 6 \cdot C6 6 + C1 6 \cdot C5 6 + . . . + C6 6 \cdot C0 6 = \dfrac{12 !}{6 ! \cdot 6 !}$

$⇒ C0 6 \cdot C6 6 + C1 6 \cdot C5 6 + . . . + C6 6 \cdot C0 6 = \dfrac{12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 !}{6 \times 5 \times 4 \times 3 \times 2 \times 1 \times 6 !}$

$⇒ C0 6 \cdot C6 6 + C1 6 \cdot C5 6 + . . . + C6 6 \cdot C0 6 = 924 .$