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Your Class 11 maths book says that Pascal's triangle "is also known as Meru Prastara". The person who wrote down how to build it, in words that survive, was the 10th-century commentator Halayudha. This post shows his construction, explains what the numbers count, works through two uses you will meet in JEE, and sets out the honest debate about how far back the idea goes. Our earlier post on Pingala's metres and binary-style counting covers how the patterns themselves are listed; here the focus is the triangle.
Who was Halayudha?
Halayudha wrote the Mrtasanjivani, a commentary on Pingala's Chandahshastra, the ancient treatise on Sanskrit metre. Jayant Shah of Northeastern University, in A History of Pingala's Combinatorics, dates the commentary to about 950 CE. Pingala's own sutras are extremely short, so most readers have understood them through Halayudha.
How did Halayudha build the Meru?
Shah translates his instructions:
"First write one square cell at the top. Below it, write two cells, extending half way on both sides. Below that three, below that again, four, until desired number of places (are obtained.) Begin by writing 1 in the first cell. In the other cells, put down the sum of the numbers of the two cells above it."
Meru is the mythical mountain, and the stacked cells look like one. The first rows are:
1
1, 1
1, 2, 1
1, 3, 3, 1
1, 4, 6, 4, 1
1, 5, 10, 10, 5, 1
1, 6, 15, 20, 15, 6, 1
What do the numbers count?
A metre is a string of heavy (G) and light (L) syllables. Shah explains that the row for n syllables gives how many forms have 0, 1, 2, ... light syllables. In modern notation the entries are nCk.
Check the row for 4 syllables, 1, 4, 6, 4, 1. The forms with exactly two Ls are LLGG, LGLG, LGGL, GLLG, GLGL and GGLL: six of them. The row adds up to 1 + 4 + 6 + 4 + 1 = 16 = 2⁴, every possible 4-syllable pattern.
The addition rule works because of a simple choice. A pattern of n syllables with k Ls either ends in G, leaving k Ls among the first n − 1 syllables, or ends in L, leaving k − 1. So nCk = (n−1)Ck + (n−1)C(k−1), the two cells above.
Worked example: using row 6 and row 5
Counting. How many 6-syllable metres have exactly 2 light syllables? Row 6 is 1, 6, 15, 20, 15, 6, 1, and the entry for k = 2 is 15. Check: 6C2 = (6 × 5)/2 = 15. The row sums to 64 = 2⁶.
Binomial expansion. The NCERT Class 11 chapter on the binomial theorem expands (2x + 3y)⁵ with row 5, 1, 5, 10, 10, 5, 1. Take the term in x³y²: 10 × (2x)³ × (3y)² = 10 × 8 × 9 x³y² = 720x³y², which matches the book's answer.
Did Pingala know the triangle?
Here the history needs care. Halayudha read Pingala's last sutra, "pare purnam iti", as the instruction to build the Meru. Shah reports that the 19th-century scholar Albrecht Weber rejected this reading, and that Alsdorf defended Halayudha in 1933. Shah himself agrees with Weber that the sutra does not say this, but argues that Pingala's computation of nCk can still be traced to a different sutra. He also notes that Virahanka, in the 7th century, described two ways of building the same triangle, which he called the needle and the mountain.
The NCERT chapter's historical note credits the Meru Prastara to Pingala's Chhanda Shastra and notes that the same triangle appears in Chinese mathematics in 1303 and in Pascal's work in the 17th century. The fair summary is that Indian prosodists computed these numbers very early, and that Halayudha gave the clearest surviving instructions for the triangle.
Which mistakes does this help you avoid?
Starting the row count at 1. (a + b)⁵ uses the row that begins 1, 5.
Forgetting symmetry. nCk = nC(n−k), so 6C4 = 6C2 = 15. Use it to check your row.
Losing coefficients inside the brackets. In (2x + 3y)⁵, the triangle gives 10, but the term also needs 2³ and 3².
For practice, see our guides to the binomial theorem for JEE and to permutations and combinations.
What can a student learn from Halayudha?
A commentator can add as much as an author. Halayudha turned a terse rule into a picture anyone could build, and that picture outlived the debate about what the original meant. When a chapter's formula feels opaque, building a table by hand, as he did, often makes the pattern obvious.
If nCk still feels like symbols, the Fermi Concept Library covers every concept in mathematics, so you can revisit combinations before returning to the binomial theorem.
Try it now: build the Meru to row 8 and use it to find the coefficient of x³ in (1 + x)⁸.
