Question

use pascals triangle to complete the expansion of (s + t)^5. s^5 + 5s^4t + 10s^3t□ + 10s^2t^3 + 5s□t^4 + t^5

Explanation:

Step1: Recall binomial expansion pattern

In the binomial expansion of \((a + b)^n\), the exponents of \(a\) decrease by 1 and exponents of \(b\) increase by 1 in each term. For \((s + t)^5\), the general term is \(C(5, k)s^{5 - k}t^{k}\) where \(k\) starts from 0.

Step2: Analyze the first blank term

The term is \(10s^{3}t^{\square}\). Here, the exponent of \(s\) is 3, so the exponent of \(t\) should be \(5 - 3=2\)? Wait, no, let's check the pattern. The first term is \(s^5t^0\), second \(s^4t^1\), third should be \(s^3t^2\) (since exponent of \(t\) increases by 1 each time: 0,1,2,3,4,5). So the first blank (exponent of \(t\)) is 2.

Step3: Analyze the second blank term

The term is \(5s^{\square}t^4\). The exponent of \(t\) is 4, so the exponent of \(s\) should be \(5 - 4 = 1\) (since \(s^{5 - k}t^{k}\), here \(k = 4\), so \(5 - 4=1\)).

Answer:

First blank: 2, Second blank: 1

So the filled terms are \(10s^{3}t^{2}\) and \(5s^{1}t^{4}\)