38:T1957,

- {}^{15}{C_1} + 2.{}^{15}{C_2} - 3.{}^{15}{C_3} + ....\,. - 15.{}^{15}{C_{15}}

29:["$","div",null,{"className":"min-w-0 max-w-[68ch] flex-1 text-[15px] leading-relaxed text-ink-600 [&_.katex-display]:my-3 [&_.katex-display]:overflow-x-auto","dangerouslySetInnerHTML":{"__html":"$38"}}] 39:Tf1f,

= \sum\limits_{r = 1}^{15} {{{( - 1)}^r}.r.{}^{15}{C_r}}

2a:["$","li","1",{"className":"flex items-start gap-3","children":[["$","span",null,{"className":"mt-0.5 flex h-5 w-5 shrink-0 items-center justify-center rounded-md bg-surface-sunken font-mono text-[11px] text-ink-400","children":2}],["$","div",null,{"className":"min-w-0 max-w-[68ch] flex-1 text-[15px] leading-relaxed text-ink-600 [&_.katex-display]:my-3 [&_.katex-display]:overflow-x-auto","dangerouslySetInnerHTML":{"__html":"$39"}}]]}] 3a:T135a,

= \sum\limits_{r = 1}^{15} {{{( - 1)}^2}.r.{{15} \over r}} .{}^{14}{C_{r - 1}}

2b:["$","li","2",{"className":"flex items-start gap-3","children":[["$","span",null,{"className":"mt-0.5 flex h-5 w-5 shrink-0 items-center justify-center rounded-md bg-surface-sunken font-mono text-[11px] text-ink-400","children":3}],["$","div",null,{"className":"min-w-0 max-w-[68ch] flex-1 text-[15px] leading-relaxed text-ink-600 [&_.katex-display]:my-3 [&_.katex-display]:overflow-x-auto","dangerouslySetInnerHTML":{"__html":"$3a"}}]]}] 3b:Tf5f,

= 15\sum\limits_{r = 1}^{15} {{{( - 1)}^2}.{}^{14}{C_{r - 1}}}

2c:["$","li","3",{"className":"flex items-start gap-3","children":[["$","span",null,{"className":"mt-0.5 flex h-5 w-5 shrink-0 items-center justify-center rounded-md bg-surface-sunken font-mono text-[11px] text-ink-400","children":4}],["$","div",null,{"className":"min-w-0 max-w-[68ch] flex-1 text-[15px] leading-relaxed text-ink-600 [&_.katex-display]:my-3 [&_.katex-display]:overflow-x-auto","dangerouslySetInnerHTML":{"__html":"$3b"}}]]}] 3c:T18fa,

= 15\left( { - {}^{14}{C_0} + {}^{14}{C_1} - {}^{14}{C_2} + .... - {}^{14}{C_{14}}} \right)

2d:["$","li","4",{"className":"flex items-start gap-3","children":[["$","span",null,{"className":"mt-0.5 flex h-5 w-5 shrink-0 items-center justify-center rounded-md bg-surface-sunken font-mono text-[11px] text-ink-400","children":5}],["$","div",null,{"className":"min-w-0 max-w-[68ch] flex-1 text-[15px] leading-relaxed text-ink-600 [&_.katex-display]:my-3 [&_.katex-display]:overflow-x-auto","dangerouslySetInnerHTML":{"__html":"$3c"}}]]}] 3d:T43b,

= 15(0) = 0

2e:["$","li","5",{"className":"flex items-start gap-3","children":[["$","span",null,{"className":"mt-0.5 flex h-5 w-5 shrink-0 items-center justify-center rounded-md bg-surface-sunken font-mono text-[11px] text-ink-400","children":6}],["$","div",null,{"className":"min-w-0 max-w-[68ch] flex-1 text-[15px] leading-relaxed text-ink-600 [&_.katex-display]:my-3 [&_.katex-display]:overflow-x-auto","dangerouslySetInnerHTML":{"__html":"$3d"}}]]}] 2f:["$","li","6",{"className":"flex items-start gap-3","children":[["$","span",null,{"className":"mt-0.5 flex h-5 w-5 shrink-0 items-center justify-center rounded-md bg-surface-sunken font-mono text-[11px] text-ink-400","children":7}],["$","div",null,{"className":"min-w-0 max-w-[68ch] flex-1 text-[15px] leading-relaxed text-ink-600 [&_.katex-display]:my-3 [&_.katex-display]:overflow-x-auto","dangerouslySetInnerHTML":{"__html":"We know,"}}]]}] 3e:T1993,

\Rightarrow

2^14 - 1

= {}^{14}{C_1} + {}^{14}{C_3} + {}^{14}{C_5} .... + {}^{14}{C_{13}}

30:["$","li","7",{"className":"flex items-start gap-3","children":[["$","span",null,{"className":"mt-0.5 flex h-5 w-5 shrink-0 items-center justify-center rounded-md bg-surface-sunken font-mono text-[11px] text-ink-400","children":8}],["$","div",null,{"className":"min-w-0 max-w-[68ch] flex-1 text-[15px] leading-relaxed text-ink-600 [&_.katex-display]:my-3 [&_.katex-display]:overflow-x-auto","dangerouslySetInnerHTML":{"__html":"$3e"}}]]}] 3f:T1990,

\Rightarrow

2¹³

= {}^{14}{C_1} + {}^{14}{C_3} + {}^{14}{C_5} .... + {}^{14}{C_{13}}

31:["$","li","8",{"className":"flex items-start gap-3","children":[["$","span",null,{"className":"mt-0.5 flex h-5 w-5 shrink-0 items-center justify-center rounded-md bg-surface-sunken font-mono text-[11px] text-ink-400","children":9}],["$","div",null,{"className":"min-w-0 max-w-[68ch] flex-1 text-[15px] leading-relaxed text-ink-600 [&_.katex-display]:my-3 [&_.katex-display]:overflow-x-auto","dangerouslySetInnerHTML":{"__html":"$3f"}}]]}] 32:["$","li","9",{"className":"flex items-start gap-3","children":[["$","span",null,{"className":"mt-0.5 flex h-5 w-5 shrink-0 items-center justify-center rounded-md bg-surface-sunken font-mono text-[11px] text-ink-400","children":10}],["$","div",null,{"className":"min-w-0 max-w-[68ch] flex-1 text-[15px] leading-relaxed text-ink-600 [&_.katex-display]:my-3 [&_.katex-display]:overflow-x-auto","dangerouslySetInnerHTML":{"__html":"Also let, S = ¹⁴C₁ + ¹⁴C₃ + ¹⁴C₅ + ...+ ¹⁴C₁₁"}}]]}] 33:["$","li","10",{"className":"flex items-start gap-3","children":[["$","span",null,{"className":"mt-0.5 flex h-5 w-5 shrink-0 items-center justify-center rounded-md bg-surface-sunken font-mono text-[11px] text-ink-400","children":11}],["$","div",null,{"className":"min-w-0 max-w-[68ch] flex-1 text-[15px] leading-relaxed text-ink-600 [&_.katex-display]:my-3 [&_.katex-display]:overflow-x-auto","dangerouslySetInnerHTML":{"__html":"

\\Rightarrow

S + ¹⁴C₁₃ = ¹⁴C₁ + ¹⁴C₃ + ¹⁴C₅ + ...+ ¹⁴C₁₁ + ¹⁴C₁₃"}}]]}] 34:["$","li","11",{"className":"flex items-start gap-3","children":[["$","span",null,{"className":"mt-0.5 flex h-5 w-5 shrink-0 items-center justify-center rounded-md bg-surface-sunken font-mono text-[11px] text-ink-400","children":12}],["$","div",null,{"className":"min-w-0 max-w-[68ch] flex-1 text-[15px] leading-relaxed text-ink-600 [&_.katex-display]:my-3 [&_.katex-display]:overflow-x-auto","dangerouslySetInnerHTML":{"__html":"

\\Rightarrow

S + ¹⁴C₁₃ = 2¹³"}}]]}] 35:["$","footer",null,{"className":"flex flex-wrap gap-2 border-t border-rule pt-4 text-[13px] text-ink-400","children":["JEE Main 2021 · 24 Feb, Morning shift · Q2991",["$","span",null,{"className":"text-rule-strong","children":"·"}],"Binomial Theorem"]}] 36:["$","section",null,{"className":"flex flex-col gap-3 border-t border-rule pt-6","children":[["$","h2",null,{"className":"m-0 text-[12px] font-semibold uppercase tracking-[0.06em] text-ink-400","children":["More on ","Binomial Theorem"]}],["$","ul",null,{"className":"m-0 flex list-none flex-col gap-2 p-0","children":[["$","li","d828964a-ded2-4fa7-868b-eecc1a0e7fea",{"children":["$","$L8",null,{"href":"/q/the-positive-integer-just-greater-than-left-5d6Ld5edVvT7r0SH","className":"flex flex-col gap-0.5 rounded-xl bg-surface-card px-4 py-3 no-underline shadow-soft transition-shadow hover:shadow-lift","children":[["$","span",null,{"className":"text-[15px] font-bold text-ink-900","children":"Binomial Theorem"}],["$","span",null,{"className":"text-[13px] text-ink-400","children":"JEE Main 2002 · Annual, Paper 1 · Q2886"}]]}]}],["$","li","e3f3c78a-ddc9-49d2-8063-e4c9f6c5e579",{"children":["$","$L8",null,{"href":"/q/the-lowest-integer-which-is-greater-than-l-1krzmg688","className":"flex flex-col gap-0.5 rounded-xl bg-surface-card px-4 py-3 no-underline shadow-soft transition-shadow hover:shadow-lift","children":[["$","span",null,{"className":"text-[15px] font-bold text-ink-900","children":"Binomial Theorem"}],["$","span",null,{"className":"text-[13px] text-ink-400","children":"JEE Main 2021 · 25 Jul, Evening shift · Q2887"}]]}]}],["$","li","bc5fc477-7ed9-4f43-a5ca-b156e69a7599",{"children":["$","$L8",null,{"href":"/q/let-and-be-the-coefficients-of-seventh-a-lsamwj6p","className":"flex flex-col gap-0.5 rounded-xl bg-surface-card px-4 py-3 no-underline shadow-soft transition-shadow hover:shadow-lift","children":[["$","span",null,{"className":"text-[15px] font-bold text-ink-900","children":"Binomial Theorem"}],["$","span",null,{"className":"text-[13px] text-ink-400","children":"JEE Main 2024 · 1 Feb, Evening shift · Q2888"}]]}]}],["$","li","875f3415-37e0-4201-9ea9-6c4f2dce13b9",{"children":["$","$L8",null,{"href":"/q/the-sum-of-all-rational-terms-in-the-expansion-o-lv0vxd0y","className":"flex flex-col gap-0.5 rounded-xl bg-surface-card px-4 py-3 no-underline shadow-soft transition-shadow hover:shadow-lift","children":[["$","span",null,{"className":"text-[15px] font-bold text-ink-900","children":"Binomial Theorem"}],["$","span",null,{"className":"text-[13px] text-ink-400","children":"JEE Main 2024 · 4 Apr, Morning shift · Q2889"}]]}]}]]}]]}]

More on Binomial Theorem

More on Binomial Theorem