Home › CAT › quantitativeaptitude › Binomial Theorem › Use the binomial theorem to compute $(101)^4$.
Use the binomial theorem to compute $(101)^4$.
A$104060401$
B$104040601$
C$101040401$
D$104021601$
Answer & Solution
Correct answer: A. $104060401$
1. Write $(101)^4 = (100 + 1)^4$ and expand using the binomial theorem.
2. $(100+1)^4 = \binom{4}{0}\,100^4 + \binom{4}{1}\,100^3 + \binom{4}{2}\,100^2 + \binom{4}{3}\,100 + \binom{4}{4}$.
3. Plug in binomial coefficients $1, 4, 6, 4, 1$:
- $1 \cdot 100\,000\,000 = 100{,}000{,}000$
- $4 \cdot 1\,000\,000 = 4{,}000{,}000$
- $6 \cdot 10\,000 = 60{,}000$
- $4 \cdot 100 = 400$
- $1 \cdot 1 = 1$
4. Sum: $100{,}000{,}000 + 4{,}000{,}000 + 60{,}000 + 400 + 1 = 104{,}060{,}401$.
5. Other options are arithmetic slips (option B swaps two digits, C drops a millions slot, D wrong coefficients).
_Source: NCERT Class 11 Mathematics, Ch 7, Example 3 (Similar method), p. 7._
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