Question

use pascals tr
(np - 1)^4 =

Explanation:

To expand \((np - 1)^4\) using Pascal's Triangle, we first recall that the coefficients for the 4th power (since the exponent is 4) in Pascal's Triangle are \(1, 4, 6, 4, 1\). The binomial expansion formula is \((a + b)^n=\sum_{k = 0}^{n}\binom{n}{k}a^{n - k}b^{k}\). Here, \(a=np\), \(b=- 1\) and \(n = 4\).

Step 1: Identify the terms of the expansion

Using the coefficients from Pascal's Triangle and the binomial expansion formula:

• When \(k = 0\): \(\binom{4}{0}(np)^{4-0}(-1)^{0}=1\times(np)^{4}\times1=n^{4}p^{4}\)
• When \(k = 1\): \(\binom{4}{1}(np)^{4 - 1}(-1)^{1}=4\times(np)^{3}\times(-1)=-4n^{3}p^{3}\)
• When \(k = 2\): \(\binom{4}{2}(np)^{4-2}(-1)^{2}=6\times(np)^{2}\times1 = 6n^{2}p^{2}\)
• When \(k=3\): \(\binom{4}{3}(np)^{4 - 3}(-1)^{3}=4\times(np)^{1}\times(-1)=-4np\)
• When \(k = 4\): \(\binom{4}{4}(np)^{4-4}(-1)^{4}=1\times1\times1 = 1\)

Step 2: Combine the terms

Adding up all the terms from the expansion:

$$
LATEXBLOCK0
$$

Final Answer

\(\boxed{n^{4}p^{4}-4n^{3}p^{3}+6n^{2}p^{2}-4np + 1}\)

Answer:

To expand \((np - 1)^4\) using Pascal's Triangle, we first recall that the coefficients for the 4th power (since the exponent is 4) in Pascal's Triangle are \(1, 4, 6, 4, 1\). The binomial expansion formula is \((a + b)^n=\sum_{k = 0}^{n}\binom{n}{k}a^{n - k}b^{k}\). Here, \(a=np\), \(b=- 1\) and \(n = 4\).

Step 1: Identify the terms of the expansion

Using the coefficients from Pascal's Triangle and the binomial expansion formula:

• When \(k = 0\): \(\binom{4}{0}(np)^{4-0}(-1)^{0}=1\times(np)^{4}\times1=n^{4}p^{4}\)
• When \(k = 1\): \(\binom{4}{1}(np)^{4 - 1}(-1)^{1}=4\times(np)^{3}\times(-1)=-4n^{3}p^{3}\)
• When \(k = 2\): \(\binom{4}{2}(np)^{4-2}(-1)^{2}=6\times(np)^{2}\times1 = 6n^{2}p^{2}\)
• When \(k=3\): \(\binom{4}{3}(np)^{4 - 3}(-1)^{3}=4\times(np)^{1}\times(-1)=-4np\)
• When \(k = 4\): \(\binom{4}{4}(np)^{4-4}(-1)^{4}=1\times1\times1 = 1\)

Step 2: Combine the terms

Adding up all the terms from the expansion:

$$
LATEXBLOCK0
$$

Final Answer

\(\boxed{n^{4}p^{4}-4n^{3}p^{3}+6n^{2}p^{2}-4np + 1}\)