Let λ denote the number of terms in the expansion of 1 + 5 x + 10 x 2 + 10 x 3 + 5 x 4 + x 5 20 . If unit’s place and ten’s place digits in 3 λ are O and T , then O + T is equal to

Question

Let λ denote the number of terms in the expansion of 1 + 5 x + 10 x 2 + 10 x 3 + 5 x 4 + x 5 20 . If unit’s place and ten’s place digits in 3 λ are O and T , then O + T is equal to,

MARKS App · Accepted Answer

1 + 5 x + 10 x 2 + 10 x 3 + 5 x 4 + x 5 20 = 1 + x 5 20 = 1 + x 100 Number of terms = 101 = λ 3 101 = 3 ⋅ 3 100 = 3 ⋅ 9 50 = 3 10 - 1 50 = 3 C 50 0 ⋅ 10 50 - C 50 1 ⋅ 10 49 + . … … . . + C 50 48 ⋅ 10 2 - C 50 49 ⋅ 10 + C 50 50 = 3 100 K + 50 × 49 2 × 100 - 50 × 10 + 1 = 3 K × 100 + 3 × 25 × 49 × 100 - 500 × 3 + 3 = Multiple of 100 + 3 ⇒ one’s place is 3 and ten’s place is 0 Hence, O + T = 3

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